Sign in
The Contribute button on the site, and the email fallback.
From the site
Sign in to the site as a member, as you normally do. If your tier is one the site sells contributions to, a Contribute button appears (floating, or wherever the site's theme placed it). Press it: a new tab opens on the site's contributor dashboard with you signed in. No password, no new account: the site vouches for you.
The dashboard lives at https://<the site's name>.<our domain>/, which the site's owner chose; bookmark it.
By email
If the button does not appear for you, or you prefer it, open the dashboard's address directly and enter the email address of your membership. A sign-in email arrives if that address belongs to a member who may contribute; the page never says whether it does. The email carries a link and a six-digit code: open the link and press its button, or type the code on the page you asked from. Both work once and for 15 minutes, and a chat app or mail scanner that opens the link cannot spend it for you.
Why nothing happens
The button only appears for a signed-in member of a mapped tier. If you are a free member, or your subscription lapsed, the site's landing page on SponsorPass shows the tier to join. If you are a member of the right tier and still see nothing, the site's owner can check the mapping; the email sign-in works regardless.