The only restriction is that a character can't play the same mission twice. A player could play the same mission three times with three different characters, though. It takes a bit of the surprise out of it, unless you have the time as a GM to work in little surprises, so those players should avoid using their player knowledge to spoil the run for those who haven't played it. Sadly, the Welcome events don't have logs or guidelines for tracking a character into the Missions campaign, so they can't take any Welcome rewards (whether karma, nuyen, or other) and remain Missions-legal. (I've thought a half-mission worth of rewards might be appropriate, given there's usually about a half-session of play.)
If you're expecting a lot of repeat players/characters, you could put together a Google spreadsheet to track what characters have played what missions. That works reasonably well with my group, as long as players are conscientious about inputting (or letting you know) when they've done a mission (or if your table is the only one they're playing at). Our team tracker just has a column for each character (under the player's name), a row for each mission, and an X if they've done it and a N/A if they won't or can't play that mission for some reason.