"Daoism" as something which people self-identified with seems to first exist around the 5th century and was a response to the growing power of Buddhism. See the following on page 16 in Kirkland's work:
"The first socio-cultural group whose participants consciously identified themselves as 'Taoist' - and began conceiving the first comprehensive collection of Taoist texts - appeared in what some would call 'early Medieval China,' during the fifth century CE. That group consisted specifically of people whose sense of Taoist identity was stimulated by the fact that Buddhism had gained acceptance and political favor throughout the land, which was, at that time, politically divided, with one imperial regime in the north and another in the south. There were many then, in the north and south alike, who had no wish to identify themselves with Buddhism."
See Taoism: The Enduring Tradition by Kirkland, a preview of which is available on Google Books:
If we accept Kirkland's assertion then clearly Buddhism in China pre-dates the formation of a "Daoism" which people can be found self-identifying with. Buddhism in China is visibly present in the histories starting in the Eastern Han dynasty 東漢 (25-220 CE), which as the Hou Han Shu 《後漢書 》 (the history book detailing said dynasty) records became quite popular. See the following:
"It has been passed down through the generations that Emperor Ming had seen a golden man in a dream who was big and tall with a halo atop his head. He asked his ministers about this. One suggested, 'In the west there is a spirit named Buddha. His figure is one zhang and six chi tall and his colour is golden yellow.' The Emperor in response to this dispatched a delegation to India to ask of the Buddha's way and methods and they succeeded in bringing images and sculptures back to China. Chu Wang Ying started to have conviction in the methods [of Buddhism]. It is because of this that China has many who revere the path [of the Buddha]. ..."
In the later Han both Laozi and the Buddha as idols were worshipped alongside each other. During the Han Dynasty they seem to have been incorporated into or at least worshipped alongside occult practices, specifically Chen Wei 讖緯, a kind of divination practice. At one point in the dynasty it became treasonous to practice it and curiously mention of the Buddha likewise disappears for about a century, which perhaps indicates worship of the Buddha was closely linked to it. Again, despite the image of the Buddha and perhaps some basic Buddhist doctrine being absorbed into the nebulous spiritual practices of this time period, it would be erroneous to claim Buddhism was related to anything called Daoism at the time.
In the early days a lot of Buddhist terms were translated using terminology from the contemporary philosophical lexicon, much of which was derived from texts which later individuals identifying as "Daoist" (intentionally in contrast to Buddhists) would claim as their own. This is why you hear scholars speak of "Daoist influences on early Buddhism", which is really just anachronistic and sloppy. The terminology they employed was from the common vocabulary available to the natives.
In the later Han Dynasty and subsequent kingdoms which arose following its collapse Buddhism was further introduced and some native Chinese authors were often at a loss on how to translate and interpret terminology, so they used what they had using a exegetical method called "matching terms" (Chn. ge yi 格義) with mixed results. For example the term wu-wei 無爲, originally derived from classical Chinese philosophies (none of which were specifically "Daoist" until long after their original authors had turned to dust), was used for nirvāṇa. Ultimately this was abandoned and a phonetic transcription of nirvāṇa came to be favoured, which is still the case today.
So, there was some degree of influence from contemporary philosophy of the time on Buddhism in the early centuries in China. This cannot be denied. Language influences and modifies the meaning of new words and ideas as they are absorbed.
It is quite similar to the west where Buddhism has continually and still is read through the lens of western psychology. A lot of Buddhist vocabulary in English is translated using items from the psychological lexicon for lack of immediate alternatives, or so some would suggest. For example, translating ātman as ego. This no doubt influences the way the concept is formulated in the minds of people already directly or indirectly influenced by western psychology.
The thing to keep in mind is that when we speak of "Daoist influences" it can be misleading because the texts self-identifying Daoists would claim as their own are often actually part of the shared common literary heritage of China. Zongmi might have read Laozi extensively, but then so did everyone else. Some might have claimed the text as their holy scripture and interpreted it in their own way, but that does not mean everyone else shared the same views. Laozi was canonical to later Daoists, but reading the Daodejing 道德經 was part of anyone's standard education.
Likewise, China had its own philosophical lexicon and physics which might be called "Daoist" by some nowadays, but in reality was just the default base of knowledge for society. Yin Yang theory might be erroneously associated strictly with Daoism, but it was just as natural for a Chinese Buddhist to speak of Yin Yang as it is for me as a modern western Buddhist to speak of gravity without having to be identified as a physicist. Furthermore, Yin Yang and the Five Elements theories existed in pre-Qin times as indigenous concepts not developed from Laozi or Zhuangzi.
As to influences from Tang Dynasty Daoism, which had matured and become quite institutionalized with a canon of scripture and organized hierarchy, indeed being a powerful religious institution in its own right, on the formation of Chan or any other school of Buddhism in China, this is something I do not sense. To be clear, the Daoists of the time developed their own scriptures by plagiarizing large amounts of Buddhist works and they had their own unique pantheon as well. You simply do not see their pantheon or scriptural citations of said texts in Tang Dynasty Buddhist literature. We can imagine Buddhist authors reading their works and being aware of their pantheon of course, though if someone were to suggest a strong Daoist influence on Buddhism in this period you would have to find many citations or allusions to specifically Daoist canonical works in Buddhist writings, but this is not to be found.
I am aware that some Buddhist practitioners in the Tang Dynasty engaged in longevity practices because they thought it was the "dharma ending age" and thus liberation being impossible now they figured they would try to live until Maitreya arrived. However, that was probably a kind of fringe cult. This might be an example of Buddhists making use of Daoist longevity practices, but the extent of that influence is from my perspective minimal.
There was a movement in China, especially after the Song Dynasty it seems, that proposed the "unity of the three teachings" (Chn. san jiao he yi 三教合一), which was in vogue not just with Neo-Confucians, but a few eminent Chan masters like Hanshan Deqing 憨山德清 (1546–1623). Indeed, this is perhaps where clear Daoist influences might be discerned when we see citations from Daoist authors being used to justify and/or affirm things in Buddhist commentary literature such as during the Ming, so clearly some influence was present in Hanshan's time.
We perhaps see the beginnings of this movement in earlier treatises such as those of the Chan and Huayan patriarch Zongmi 宗密 (780–841) expressing such sentiments in his famous work the Yuan Ren Lun 元人論 (Treatise on the Original Man). Zongmi was a unique case because he was very well learned in non-Buddhist literature before taking an interest in Buddhism and later renouncing to become a monk. Zongmi still had a taxonomy of teachings and made judgements on the superiority of some teachings in relation to others. Zongmi suggested that while the teachings of Laozi and Confucius were holy, they were provisional while the Buddha's teachings were a mix of ultimate and provisional truths. Nevertheless, the purpose and end result was said to be the same and thus all three could be practised in his mind, though Zongmi does not make use of either the Daoist pantheon or cite Daoist authors.
Looking broadly at the histories of Buddhism and Daoism in China clearly the former was existent before the latter if we assume people self-identifying as "Daoist" came to exist in the fifth century. Judging from the extent institutionalized Daoism borrowed from Buddhism and the lack of the reverse we might assume that Buddhism played a dominant role. To say Daoism did not influence Buddhism would be inappropriate, particularly once you look at Ming Dynasty Buddhist commentaries which would reflect otherwise, but that influence was never prominent enough to justify closely linking the two at least on the level of the intelligentsia. Granted, on the common ground level of religious practice China indeed has and had much syncretism with the borders between "religions" (the concept of 'religion' as understood in the west required new vocabulary to be crafted in Asia in the 19th century incidentally) rather nebulous. This is still the case today.
However, that is a discussion to be had another time.