Travel Site That Allows for Half the Money Down

you need to go to a travel agent.

We book our flights in advance (pay in full) and make hotel reservations DIRECTLY through the hotel website/phone #. The hotel only requires a credit card number to hold the reservation, they don't actually charge you anything until you arrive (just make sure you're aware of the hotel's cancellation policy, usually you can cancel/make changes up to 24 hours prior to your arrival.)
We travel a LOT and have noticed that the hotel rates on Expedia or Hotels.com or whatever really aren't any cheaper than booking with the hotel directly, and often you have to pay in full when booking and it's non refundable so it's really not worth it.

Airlines, hotels and rental car agencies do not work that way.

Airlines sell tickets based on the projected seating and you pay in full at the time of booking to hold your seat.

Hotels and rental car agencies will hold your credit card and run the authorization when you check in OR charge you if you do not show up and do not cancel within the timeframe mentioned in the fine print.

If you choose to go through a discounter, be prepared for potential issues because if any airline, hotel or car agency needs to "bump" someone, the discounters are the first ones bumped. I have seen that happen several times.

I have been booking flights weekly over 20 years. We fly a lot on business. I only deal with my specific airline, hotel and car website.

Have you spoke to a travel agent? Mine only required half down, full payment of travel insurance if you want that. Then the balance was due a month before the trip.

Beach rentals through real estate companies do that all of the time. Balance is due 30 days prior to arrival.

We found a beach rental through tripadvisor where we only had to pay a percentage upfront but it was a private house rental, not a hotel, and we didn't book a flight or car rental so I don't know if that site would work for you.

I think you'll need to go through travel agency for that.