I'm looking at the number of new entrants into an industry as a
measure of the vitality of the industry--is there a service that
tracks new entrants (may or may not be new companies) to an industry
over time? Specific industry is banking, retail banking specifically. Thanks.

Since there are state and federal banks as well as savings and loans, I doubt
any single government office would track the whole.

But they might have something representative.

Since banks tend to gobble each other up,  You might want track those
disappearing as well.

Getting authorized to start a bank is pretty tricky
- especially in the USA and Europe

I would anticipate the number of 'new entrants' as about zero

As RedFox pointed out, banks tend to merge or get bought out.

Also, I would not measure the vitality of a sector by the number of
new entrants, a consolidating and/or mutating sector could be pretty
active.

Something like banking or the steel industry could go through a boom
with no new startups, because the cost of entry is extremely high.

Banking is a conservative industry with closely interrelated services
between banks.  Even a fantastically good new vendor would have to
lock in on the existing system.  The odds that such a great idea would
result in establishment of a new bank is minimal.  Much more likely is
that the idea would be sold to an existing bank, and even more likely
is that it would have been developed by an existing bank.  Most of the
people who know the industry are employed by a bank.