How does one get Admin access to one's ExpressionEngine site?

While researching why the addition of a CAPTCHA field to our non-profit organisation’s website’s contact form seems to work in the FreeForm composer (but the actual form visitors see remains unchanged), I stumbled across some possible solution paths that involve going into the admin section of Core / ExpressionEngine. We only have one username and password for our ExpressionEngine, and while logged in as that user, we get a not authorised response when we click on either Admin, or Admin->Overview, or Admin->Software Registration.

It looks like this when we try: https://vimeo.com/328469801/97c9a09f17

So… not sure how to go about getting Admin access so that we can troubleshoot why CAPTCHAs aren’t working. Instead of troubleshooting Captcha at all, a solution proposed in the solspace-tagged threads at stackexchange was to purchase Snaptcha, though my understanding is that to install an add-in would still require us to have admin access.

All insights into how I can go about getting said access are most welcome!

(For what it’s worth, we contacted the firm that created the site (which was many moons ago), and they offered to bill us for a few hours work (at an estimated $150-to-$250 per hour) to see if they could find whether or not there is a super-admin user for our site that could give us what we need… so since I am costing us only $27/hr to reach out for help, well, we’re trying the “Help!, COmmunity, help!” route first.