Approval for a vaccine against COVID-19 depends on a company’s application– or does it?

Phase III clinical trials of seven or eight vaccines are in progress. They come from CanSino (Ad5-nCoV, adenovirus-based– actually in limited use in China), Gamaleya (adenoviruses Ad5 and Ad26-based Sputnik V, in limited use in Russia) AstraZeneca/Oxford (AZD 1222, ChAdOx1 chimpanzee adenovirus-based), Moderna/NIH (mRNA-1273, mRNA-based), Pfizer/Fosun/BioNTech (BNT162b2, mRNA-based), SinoVac (CoronaVac, inactivated virus-based), and SinoPharm (two different, inactivated virus-based– one of them already in limited use in China.)
See this New York Times interactive article, updated September 8 and probably daily, for a timely update on the latest vaccine information.
The companies suggest that their final Phase III studies will be finished around the end of this year, but none of them promise anything before the election. The president, by contrast, has virtually promised something before November 3.
A group of five vaccine companies have come together in a statement that they will not release a vaccine until it has been shown to be safe and effective. By “effective”, they have been held to the rather low standard of a 50% reduction in infections.
The real question is whether any company will request an “Emergency Use Authorization” (EUA) before the election– because the FDA can’t offer an EUA unless a company asks for it. At least, I don’t think so. Given that, I don’t see how the president can get to his goal of announcing a vaccine before the election.
If anyone can explain to me how the president can get a vaccine company to ask for an EUA before the election, or force the EUA through without a company request, I wish they would tell me. This question really bothers me because I know the president is desperate to announce a vaccine or at least a breakthrough pharmaceutical (which does not appear to be on the horizon)– so how is he going to do this?