The predictable answer is "maybe", and you are certainly right to stop and think about the possible ethical implications of your use of the website. There are two key issues to look at to determine whether what you are doing is problematic.
First, are there any express terms of the website (such as there often are on internet auction sites, for example) forbidding you from "going private" to complete a deal that you have discovered on the internet? Although a prohibition of that kind may be virtually unenforceable in practice that does not prevent it from being binding in halachah. So if the website says anything along the lines of "please do not attempt to book privately with any hotel featured on our website" that would be enough to make it, halachically, an implied condition of your right to use the search facility on the website, and breaching that condition would bring you within the broad confines of the laws of gezel (theft).
Secondly, terms and conditions aside, you need to think broadly about how the website makes money, and apply the halachic principles that prevent one from doing anything that diminishes another person's reasonable expectation of making a livelihood (yeridat umanut).
Many websites make all or most of their money through pop-up advertising that appears while you are searching the site: in that case, the website may not be aiming to make any money by way of commission from deals made through it, in which case by contacting the hotel privately you are not causing the website's owners any loss at all. But if there is any reason to think that the website gets a commission from deals made through it, your use of the facility to find a deal but then to complete it privately would again bring you close to issues of theft in halachah, and would certainly involve reducing someone's legitimate income.
The situation is also not far from the Mishnah prohibiting one from going into a shop and asking the price of things one does not intend to buy, because it raises false expectations for the shop-keeper: if the website owners monitor traffic they may be hoping that a certain percentage of hits will result eventually in deals from which they earn commission, and if lots of people do what you do, the owners will find their expectations disappointed.
There may be some indications on the site itself that show to what extent its owners care about private deals. For example, on some sites the telephone numbers of hotels are not given, suggesting that the site owners want you to book through them. Failing that, you should probably contact the website and ask whether they mind if you use their site to make contacts and then deal direct: that may seem a very punctilious thing to do, but the Torah requires us to be punctilious in our dealings, and it is better to look a bit naive than to risk falling foul of the very broad and exacting laws of Jewish business ethics.