We don't live in Portugal yet, but got NIF numbers while in the UK and without having a Power of Attorney, I don't know how it was done mind you, the estate agent we are using managed to do it on our behalf. I had to send copies of our proof of address and passports and sign a letter of authorisation to say that we were happy for the agent to apply on our behalf. I think the cost was a little over €20 for us both. We also opened a bank account with Santander, but, although the paperwork was all in place for us to sign, we did have to visit the bank here in the UK to sign it all. I believe there are bank accounts you can open on-line though, Activo possibly may be one of them, so worth looking into.
As @x-camone says, you could most likely get an NIF number by appointing a power of attorney. We have one of those in place for our lawyer to sign the purchase documents on our behalf. Fingers crossed the promissory note exchange should be happening this week.
Hope that helps and good luck 😊