If my supplier charges me in dollars but my customer pays me in my own currency, how does the system handle the exchange rate difference?

Yes, a booking system can handle this, but you have to decide in advance how you want the conversion to work, because the difference does not disappear on its own.

In most setups the supplier sends back the rate in their own currency, the system converts it into your selling currency using either a rate you enter yourself or a rate taken from a live currency feed, and then your markup is added on top of the converted amount. Your customer only ever sees and pays in the currency you have chosen for your website. Behind the scenes the booking is still recorded in the supplier currency as well, so when you settle with that supplier you are comparing like with like.

The honest limit is timing. The rate on the day the customer books is rarely the same as the rate on the day you actually pay the supplier, so a small gain or loss is normal on every booking. Your payment gateway may also apply its own conversion when it pays out to you, and that is set by the gateway and your bank, not by the booking system. Some suppliers will only settle in their contracted currency, so you cannot choose to pay them in yours. Which currencies you can sell in, and whether live rates are available or you have to maintain them manually, depends on your gateway and your plan.

Before you switch on a second currency, ask your payment provider which currencies it will actually settle into your bank account, then decide whether you want to build a small cushion into your markup to absorb rate movement, and check that your booking reports show both the supplier amount and the amount your customer paid.