Can a flight booking API show my customers the fares in their own currency?

Yes, in most cases it can. A flight booking API will usually let you ask for fares in a currency of your choosing, and the search results come back converted so your customer sees a familiar figure instead of doing the maths themselves. You set the currency either for your whole portal or per search, so a visitor from one country can be shown one currency while a visitor from another is shown a different one.

What you need to understand is that the conversion is normally a display conversion. The airline still holds the fare in its own currency, and the rate used to convert it comes from the supplier, not from a live bank rate. That means the amount your customer sees can move slightly from one day to the next, and the amount finally settled with the airline may not match the converted figure to the last unit. Some suppliers support true multi currency selling where the ticket itself is priced and paid in the chosen currency, but that depends on the supplier agreement and the markets they are set up for.

The currency your customer pays in also has to line up with your payment gateway. If the gateway only settles in one currency, showing prices in five will still end with a charge in that one, and the customer's own bank may add a conversion charge on top. It is worth deciding early which currencies you genuinely want to sell in rather than switching them all on.

Before you promise anything to your customers, check with your supplier which currencies they support for search and for ticketing, and check what your payment provider can actually settle, then match the list on your portal to whichever of those two is shorter.