How do I contact an HSA administrator? Who do I call/email?

if HSA is through payroll, ask your payroll department. if not there must be forms that you submit to make  payments call them 

I don't work at the job with the HSA anymore. Contacting anyone specific there is nigh impossible at this point. Plan B?

Obtain from the IRS copies of your Wage and Income transcripts for recent years.  Forms 1099-SA and 5498-SA issued by the HSA custodian will be listed and will show the name of the payer who you can contact.  However, 2021 Forms 1099-SA and 5498-SA might not appear in your transcript until later this year, so you'll need to look at your 2020 Wage and Income transcript (assuming you had an HSA in 2020).

Of course if you made no distributions from the HSA in 2021 (made no medical payments from the HSA), there will be no 2021 Form 1099-SA.