To "shop" is to search for things that you may want to buy or rent. It does not necessarily imply that you are buying outright and not renting. In the U.S., anyway, people routinely talk about "shopping for an apartment", "shopping for a rental car", etc. There's a website called "shopapartments.com", where users "shop" for "apartments to rent". "shoprentone.com" is a site where users can rent (not buy) home appliances, and it has "shop" in the name as well as a "Shop Now" button. (Disclaimer: I'm not recommending either of these sites. I just found them now with a Bing search, I have no idea whether they are good companies or not.)
"Buy" or "purchase" indicates that you are taking ownership. You would not say that you are "buying a hotel room" if you mean that you are just paying to spend a night or two.
But there's no problem saying that you are "shopping" for things that you will rent. "Shop now" is fine for rental products.
Side note: You wouldn't say that you "give products on rent". You can say, "We offer products for rent", "We have products available for rent", or "We rent products". (The last is ambiguous, as both the person offering the product and the person paying for it are said to "rent" it. "I am renting a house" could mean that you own a house that you are charging someone else to live in, or that you are paying to live in a house that belongs to someone else.)