Technically, the members can vote however they want and that’s it. It’s not like a parliament (UK, Canada, Australia, Israel, etc.) where too many voting against the government triggers a no confidence and an election.
The members used to regularly split on their party’s bills. In the good old days, a lot of legislation was written and passed due to “Scratch my back, I’ll scratch yours.” It’s become less common as the political divide has become nastier and “cooperation” is seen as a dirty word. Plus, Trump was only the most recent to introduce the concept of “Primarying” where difficult members will find the influential national figures promoting someone else to get the nomination for the party against them in the primary. (Liz Cheney, anyone?) This extends to party support for elections, with advertising dollars and organizing. PACs are also sticking their nose into things to enforce adherence to dogma.
The party also has some influence with other things. IIRC, the head of the house or senate for that party can do things like allocate office space, and decide who gets to be on influential committees. (Although the speaker technically has the right to overrule the minority leader - but do so with caution or nothing gets agreed on). For example, the GOP wanted certain members on the House Jan 6 committee and the speaker said no, they would pick Cheney and Kinzinger.
A good example of this OP’s question is the senate so far with Sinema and Manchin. They knew if they said “no” the senate could not do many things, so much got held up trying to appease their demands. OTOH, they will probably face a backlash in terms of what they can ask of the party if they are no longer the decisive vote. And the Dems don’t want to piss them off too much, because if they defect to the Republicans, then McConnell becomes the leader of the senate.
When McConnell was senate leader, he could delay a lot of things because the senate leader decided what made it to the floor to be voted on - set the agenda. If he decided, for example, “we’re not going to vote on Garland as SCOTUS nominee”, it didn’t happen. He delayed on a large number of Obama judge nominees, thus ensuring that there were plenty of openings for federal judges when Trump got into power and nominated judges Mitch approved of.
OTOH, the senate has an interesting fake :filibuster" where 40% of the senate (40 members) can block any agenda item. (There are exceptions, like judges and budget bills).