At C2 in the English Vocabulary Profile, ‘would sooner’ means ‘would prefer.’
For example:
They would sooner sacrifice numbers to save the people.
The most common collocate or grammar structure related here is comparative ‘than’. For example:
I would sooner resign than be forced to get the vaccine.
A search in the NOW corpus for which infinitives are found next to ‘would sooner’
would sooner _VVI
1 SEE 51
2 DIE 37
3 FORGET 22
4 GO 21
5 BUY 20
6 GIVE 16
7 BELIEVE 12
8 CUT 12
9 TAKE 12
10 LEAVE 11
11 PLAY 11
12 PUT 11
13 KILL 8
14 LOSE 8
15 RUN 8
16 SPEND 8
17 JUMP 7
18 LIVE 7
19 PAY 7
20 VOTE 7
A Donald Trump supporter says she would sooner vote for Vladimir Putin in an election than Joe Biden.