Checking for vacancies on booking.com may not be comprehensive---it will not produce a list of ALL hotels with rooms available----but it definitely is reliable. They will show a list of hotels, BnB's, etc. with rooms available and the booking policy---whether prepayment is required or not. Often the same property will have both---a low prepaid price and a slightly higher "pay later" price whichnis what Kira is looking for. And you can filter results by neighborhood, price range, and other criteria.
It is a great timesaver. Kira is leaving for London in two days and probably does not want to spend half a day or more checking individual websites based on recommendations received here. She asked specifically about third-party websites that allow one to check for credit card reservation policies. I suggested booking.com as I have used them in the way I described---check there then book direct. I suppose Orbitz and Travelocity would work too but I have no experience with them---I like the functionality of the booking.com website.
Even the hotel's own website may not give complete information. Some hotels, especially small family-owned ones, use booking engines or software that do not accurately reflect complete availability. Example: In 2012 I found a very nice small hotel in the Dolomites, specifically in Santa Cristina in Val Gardena, for our next visit there. I put the Dolomites into our 2013 trip itinerary between Germany and Venice. But the dates we planned (late September) were later than what the Villa Martha website showed as their operating season. I assumed they closed for their own vacation at this time, like many other hotels and restaurants in the area (some lifts close at this time as well). So I used booking.com to check for places that would be open, and lo and behold! there was Villa Martha with the dates we wanted showing as available. So I emailed and explained that the website would not take the booking, but I wanted to book directly rather than on booking.com. Mateo was very gracious and we made the booking for three nights. We were practically the only guests but it was every bit as nice as we had read. I would never have thought to contact them had I not seen the listing on booking.com.
I do not view booking.com as the Evil Empire like some here do. Yes, they charge the hotels a commission for the booking. So some places make only a few rooms available for booking through the third party agent, to limit the number of commissions. But they do get advertising out of it, so apparently they deem it a good business practice.