Can I work in a top firm if I did not get a summer associate position?

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"I want to be a litigator in a major market, but failed to secure a summer associate position. I'm at a top law school. Will this be a permanent thing in my record that keeps me from me being a litigator at top firm?"Absolutely not. All sorts of people don't get summer jobs and then do very well. All sorts of people end up not getting even good jobs after they graduate and move up. None of this really matters in the long run. You can still work at a smaller firm over the summer, and you should try to find a job doing that. You can get other types of positions with law firms.
The reason people don't necessarily get summer associate jobs is it's difficult when you're a new attorney and when you're a law student to walk into interviews and know how to behave. You don't know necessarily how to talk to other attorneys. You don't know how to interview. You may not have the grades you wanted. You don't know necessarily what law firms are looking for. Or you just may be competing with better people or the lottery in the firms you're talking to. I actually recommend that a lot of people that are looking for summer associate jobs, that they try to do things other than go through OCI. Because if you don't get a job through OCI, where does that leave you?
So you should also be applying maybe to firms and markets where you're not from, where you have connections. And seeing if you can get a job that way as well. But no, it's not going to be a permanent thing in your record. Getting a job in a law firm, and your legal career, is a long race. Right now, let's say you're 21 years old - I know people that are practicing that are 95. I've placed people in their eighties. So you can definitely still get a job.