To clarify, your HSA contributions are not reported on the 1099-SA; only your distributions are reported on the 1099-SA (i.e., money that you paid out for medical expenses). Note that if you made no distributions from your HSA in a tax year, then you won't receive a 1099-SA at all for that year.
Carol C is quite correct that you don't get a form showing your direct HSA contributions (i.e. not through your employer) until you receive the 5498-SA, which may be mailed as late as the end of May (although it could be earlier). Note that Box 2 on the 5498-SA shows the total amount of contributions made to your HSA for the tax year, both direct contributions and through your employer. Of course, the amount of HSA contributions made through your employer (if any) are indicated by the code W amount in box 12 on your W-2.
Note that the 1099-SA will not be issued for when you made the HSA contribution, but when you make the HSA distributions. From your description, it sounds like you did the reimbursement (the HSA distribution) in early 2019; if so, you will receive the 1099-SA that includes this amount in early 2020 in time for your 2019 tax return.
One more note: ask your employer if they would consider withholding your HSA contributions from your paycheck and sending it to your HSA custodian (the credit union). The reason for this is simple: you would reduce your taxes even more. Why? When you make direct HSA contributions as you do now, you can deduct that amount in line 25 of Schedule 1 on your return. But if the HSA contributions are done through your employer (whether or not they actually contribute any themselves), the amount of the HSA contributions are removed from Wages in Box 1, 3, and 5 before your W-2 is printed. This means that you are not paying Social Security taxes nor Medicare taxes on the contributions, either. If you make direct HSA contributions, you have no way to reduce your Social Security nor Medicare taxes.
Alternatively, your employer may agree to make HSA contributions, but only to the HSA that they help you set up. That's OK, because you can have two or more HSAs. Note that all the limits apply to the aggregate of all your HSAs, so this isn't a tax dodge. And when you do reimbursements from your HSA, you can do them from either one you want (or both). In this case, ask your employer to help you set up the HSA with the institution that they prefer - note that the HSA is your account in any case (not the employer's). You will reduce your tax bill by 7.65% of your HSA contributions through your employer because you will no longer be paying Social Security and Medicare taxes on your wages. No, you won't see this on your tax return - the 1040 is the federal income tax return (not SS nor Medicare), but it's a savings nonetheless.
