In British English, a solicitor is a lawyer, but in American English that usually means someone who solicits something, which means to demand money from someone. "Endorse" can mean two things: give a signature or approve. The approval meaning is much more common. In this context, it means approve or allow. So "The airport does not endorse solicitors" essentially means "We do not allow you to sell your products here." I assume this was from an American film because I don't think Britain would be banning lawyers. haha

In the past, in the U.S., people could solicit (ask for) money in public places such as streets and pseudo-public places such as shopping malls and airports. The Salvation Army, the Girl Scouts, and other organizations would set up a table and ask for money. At a certain time in the U.S. (80s?), fringe religious groups would aggressively solicit money in airports. Airports first announced that they had no relationship with these groups and then later expelled all of them. Over time, some cities outlawed this form of money raising and nowadays most malls and similar places have signs clearly forbidding soliciting money.

solicit is a verb. It means to approach people and ask something, for example to sell something. . "unsolicited" means they approach without you asking them to come. . endorse measn to support, possibly even encourage. . The airport does not encourage of support people who approach you with offers, or asking you for something. . Sometimes "endorse" can means to check on and license. So sometimes it can be taken to mean that the airport do not vet, check, approve, license people to solicit. That is there are no approved solicitors in the airport. This is basically just a slightly stronger interpretation of a similar thing.

Stop watching Final Destination xD

Endorse could also be used to refer about agreement change. For example, if you hire a car insurance and some personal information or feature of the car was wrong in an initial agreement it is necessary endorse the agreement to correct the information.