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It looks okay. Why does it sound awkward, then:
They wanted her to leave. They did not say so openly, but many of them looked so relieved when she headed for the door that it might be implied.
I don't know ... "may have been"? "might have been"? Bummer.
First off, I agree that the tense should be the past "might have been."
But on a broader level, that whole "that it might be implied" clause seems problematic to me because it appears imprecise what "it" is referring to. What exactly is the antecedent? Of course one could say that it's the entire sentence "They wanted her to leave," but that verges on being unacceptable in my opinion to make an "it" refer to a thing so hazily. Plus "that" is located so far from "so" in the construction, which doesn't help matters. I think it would be better for one to maybe replace "it" with something like "their wish"
I think the real problem is with the word "implied," which means something has been "suggested." If "implied" is necessary, better would be: They did not say so openly, but their looks of relief implied as much when she headed for the door. Another solution is to use "infer," meaning "to conclude from evidence": but many of them looked so relieved when she headed for the door that it might have been inferred. ("Might have been" now suggests that that it was possible to infer at that time that everyone wanted her to go; but "might be inferred" also works, to mean that the inference is still there to be drawn.)
