They can cut budgets without Congress by reappropriating money now, it's one of the powers they've managed to usurp. But they don't have to cut anything, they manage to curb spending by throwing a wrench in the whole machine and watching awards crawl to a halt. They cancel grants, fire or drive out reviewers to increase review times, or delay follow-up funding. Maybe the funding comes through eventually but students need to be funded continually; the government will pull their visas if they don't have funding to enroll.

They're also straight up harassing and arresting foreign students for no reason, so they don't even have to muck with the budget at all to materially ruin things.