Governor Inslee of Washington promised to create a panel of experts to review the safety of vaccines back in October 2020. Zach Lipton states in a comment that the following concerns were up in the air during the time:

For context, keep in mind that these groups were formed during a time
when there was significant concern that the President was applying
pressure to have vaccines approved before election day for political
reasons, such as the White House attempting to shorten the time
period that clinical trial participants needed to be followed
after vaccination (the FDA ultimately stood their ground). The
President's hydroxychloroquine advocacy also caused concerns.

By the time the vaccines were ultimately authorized, this was no
longer a significant concern, and the vaccines had the unanimous
backing of well-trusted public health leaders. Furthermore, the first
groups to get vaccinated were mainly medical professionals, who lent
further credibility to the vaccines' safety. As such, these panels
didn't receive as much public attention when they announced their
results.

However its not exactly clear to me what realistic scenario existed where President Trump could've interfered into the vaccine approval process. Some plausible scenarios I could think of:

1. Trump issuing an Executive Order to approve the vaccines. This does not seem to be legally possible.
2. Pfizer trials finish and their official statement is that the vaccine is not safe - e.g. it caused 1% of participants to die. Then somehow the FDA leadership is corrupted and they still approve this vaccine. This does not look plausible, as even if the FDA approves the dangerous vaccine, you'd still have to contend with Pfizer refusing to ship out a product that would destroy their reputation. Plus it's not like you'd even need a "panel" to know taking that kind of a shot is a bad idea.
3. Trump somehow gets Pfizer to fake their trial results, which are later approved by a "corrupt" FDA. This strays far into conspiracy theory land, as the number of people who would have to keep a secret would be in the thousands.
4. Pfizer trials finish and the vaccine is only 20% effective, with a "corrupt" FDA approving it, rather than using their previous 50% threshold. This might be in the realm of possibility but once again - would Pfizer realistically agree to manufacture such an inferior vaccine en masse, knowing that other countries won't approve it?

Were there any comments in the press during the time hinting at what exact scenario was feared by Democratic leaders? Not a handwave of "well a Republican is in charge so he can do bad stuff" but rather an attempt to explain what exactly could go wrong during the process, step by step.