Renewing your Developer Program is usually just a minor annoyance. You either have it set up to auto-renew and promptly forget about it[^1] or you need to log in before the deadline for a few minutes and click on a few buttons to renew it manually. No big deal. I didn't have any problems renewing it these previous years, all the way back to 2014.
This year let's say it has been pretty complicated… I logged in and there was no buttons, after trying to find a solution on Google I came up with this crowdsourced list[^2] that solved it:
Go to your account page and look for the renewal link
Try using the direct link to the renewal page
Turn OFF auto renewal if you have it on and reload the page
Use Safari on macOS, NOT any other browser
Log out and back in (with auto-renewal turned off)
I managed to get the button showing after turning off auto-renewal, logging off and finally logging back in. All of this while using Safari.
If all else fails you can try calling Apple Support, they're usually pretty helpful.
[^1]: I would be constantly worrying if it'd have gone through or not though, cause I've heard multiple reports of it failing to renew (rarely).
[^2]: Credits at the bottom,as always.