Method for booking TUI Fly tickets from within the US?

We’ll be vacationing in Europe this summer and it turns out that only TUI Fly has a direct flight between two countries we’re visiting on days that match our schedule. Problem is that when booking the tickets on their website it craps out just before checkout (“Technical Difficulties”). Suggested solution from the web is to use an online travel agency (OTA). I’ve never booked flights anywhere other than the carriers website so I’m a little weary in case there’s an issue while traveling.

Any suggested OTA’s or other means of booking the flights?