How can I change my voicemail password if the option isn’t popping up in settings or on “phone” in settings?

Tap Phone then Voicemail.

Not sure which voicemail do you use, Carrier voicemail or Visual Voicemail messages.

Visual Voicemail messages, see this Apple article: Check your voicemail on iPhone

Check your voicemail on iPhone – Apple Support (AU)

If you use Carrier voicemail box, call your voicemail box number -> xxxx to listen to your voicemail messages.

Which voicemail box number you should call depends on your carrier. 

Contact your carrier or do a Google search.

When you call your voicemail box xxxx, listen to the main menu/options.

An example how it works with my carrier:

Call xxxx and press 11 for the main menu. Choose 2 for the personal settings of your voicemail. Choose 3 for PIN code and choose a 4 digit code. You can also change your pin code here and indicate when your voicemail should ask for the pin code.

FYI: if you enter an incorrect code 3 times and you will receive a temporary code via SMS. You can change that later.

Depending on your carrier, it may not be an option there. Try logging into your billing account for your carrier. There may be an option available.