Generally... no. You shouldn't need to give up your character's integrity. There's usually alternatives and compromises available.
Now, there is My Guy Syndrome, which is that you'll do unfun things because "it's what my guy would do", without realising you have a choice there. There's also Making the Tough Decisions which is "you can shape your character to help propel the plot". This is neither of those, this is players pressuring Player A to drop what they feel is a core part of their character and source of fun.
For some people, those core parts are important. Some players derive their joy in a game from playing out a character — it's a totally valid mode of play. They should still avoid My Guy & make tough decisions, it's not mutually exclusive. Those players are doing something unwelcome, and pressuring a fellow player to give up their source of fun and compromise their values. It's not healthy play.
So: those alternatives and compromises are how to move things forward. These players, A and B and all the others, need to sit down and talk about third possibilities beside simply "everyone including the paladin commits robbery" or "no robbery at all". There's win-win alternatives to be had if you start talking them through seriously.
Situations like the one you describe can actually create some beautiful drama to explore. There's How do I play a paladin without being a stick in the mud? to provide some basics for figuring out how to play a super-lawful paladin alongside less lawful activities, but one of the things that paladin can do is simply go along with his friends to keep them safe. They're stupid, they're doing something stupid, he hopes the robbery fails and won't steal anything himself, but he'll at least be there while they're doing it because he cares about his fellow party members and doesn't want them killed. (Does he not care about them? That might be one of those "make a tough decision" things: time to develop that the paladin kinda grudgingly cares about them, or something.)
Alternately, the paladin can at least keep watch outside for more or less the same reasons, if splitting the party is workable. He could be there to prevent the absolute worst, or stay aside to be able to vouch for them to authorities if they're caught and otherwise get them out of a mess (as RollingFeles suggests).
The paladin might not be totally happy with it, but that is a drama seed you can use for fun and character development. People get forced into unpleasant situations sometimes, or give up something for the sake of others' benefit.
(All of that considered, Player A should probably not force the party to do things their own character would not themselves do, since that would be the typical stick-in-the-mud paladin. It doesn't sound like that's happening, though.)
If Player A would feel it compromising and unwelcome to have his paladin participate in the robbery, his fellow players should respect that. Rather than asking him to quit fussing about it and go do some robbery, they should take that constraint, accept it, and discuss alternate possibilities. Player A should work out something with them that doesn't violate their core fun values, and look at compromising situations as a potential opportunity for character development and drama and its own kind of fun — but Players B onwards should respect their boundaries.