If you’re fired & it really is not your fault you have the ability to explain what happened. Interestingly, employers risk a lot if they bad-mouth an ex-employee. At the last place I worked the policy on referrals was simply to say 1) yes, they were an employee & 2) they would or would not be considered for re-hire. Nothing further would be said. I always kept my annual reviews to hand out to possible future employers.
You can try but you’ll be lucky to get a word in. I had two places go out of business within two months of each other, and it TRASHED my job search for a damn year. The few places that did give me an interview acted like I had a criminal record.
What makes it worse are the automated systems online which are often set to pre-filter anyone who checks the “I was fired” box or had more than X many jobs within Y months/years. You won’t even get told why you didn’t get any response.