Can my customers pay in their own currency while I still get paid in mine?

Yes, in most cases you can show prices and collect payment in the currency your customer is comfortable with, while your own books stay in your home currency. The booking site can display a converted price using a rate you control, and the payment gateway collects in whichever currencies you have enabled on your account.

How far this goes depends on two things that are not really in your hands. The first is your payment provider, because it has to accept the currencies you want to take and settle them into your bank account. The second is the supplier or the API behind the product, because flights and hotels are usually priced in the supplier's own currency, and that is the amount you owe them no matter what currency your customer paid you in.

Because of that gap, the safer way to run it is to keep the supplier currency as the real cost and treat the customer facing currency as a display and collection layer, with your own rate and a small buffer built into your markup. Exchange rates move between the day a customer books and the day you settle with the supplier, so without that buffer a booking that looked healthy when it was made can end up costing you by the time the supplier invoice lands.

Ask your payment provider which currencies they will accept and settle for your business, then check which currency each of your suppliers bills you in, and set your display rate and your markup so that the two always leave you covered.