As a prerequisite, get the code and database schema (+ scheduled updates to CSV's of number of rows/table, per-table storage size, per-database storage size, etc.) into source control ASAP. You can do this entirely on your own local machine day 1, automating rescanning and committing a diff at least daily. Also regularly commit all of the configuration files for the production environment (devops, installed packages/versions, bash histories, etc.) that you can get access to.

In parallel is a review of the disaster recovery plan... do a full test restore of code + data from scratch!

I would then encourage an evaluation to get the lay of the land. If my intuition is correct, there are high priority problems in production that no one is aware of, well beyond the tech debt.

Start by setting up centralized error logging as quickly as possible, from the simple 404/500 error and database timeout reporting (is there any low-hanging fruit here redirecting URLs or speeding up the DB [indexes]?) to more deeply entangled server-side error reporting... ELMAH was an eye-opener when first dropped into an existing cowboy-style ASP.NET app, I don't know if something similar exists for PHP for free but you could learn a ton just trialing a commercial APM solution (same for db optimization tools).

Then once the fires are identified and maybe even a few are out, analyze available metadata to determine the highest-traffic areas of the application. This combines client-side analytics, server-side logs, and database query profiling, and guides where issues should be fixed and tech debt should be paid down first. You can get down to "is this button clicked" if you need to, but "is this page/database table ever accessed" is helpful when getting started. (It's often nice to separate customers from employees here if you can, such as by IP if working from an office.)

Do you have the option of pursuing hardware upgrades to improve performance? (Is this on-prem?) You might want to dig into the details of the existing configuration, especially if the database hasn't been configured correctly. Which databases are on which drives/how are available iops allocated/can you upgrade RAM or SSDs? One big item here is if your are nearing any limits on disk space or iops that might mean downtime if not addressed quickly.

In the cloud you have opportunity to find resources that are not being used anymore and other ways to cut costs. Here again you can trial commercial solutions for quick wins.

Finally, implement some type of ongoing monitoring to catch anything that happens rarely but may be absolutely critical. This might be best done through an automated scan of logs for new URLs and database queries. After a year to 18 months, you should have a good picture of which portions are completely dead (and can be excised instead of fixed). You can start cutting things out much sooner than that, but don't be surprised if a show-stopping emergency comes up at the end of the fiscal year, etc.!

These are all easily justifiable actions to take as someone hired to get things headed in the right direction, and can earn the political capital necessary to begin pursuing all of the other recommendations in this thread for managing technical debt.

Edit: one mention in the thread of prioritizing restructuring the DB, sounds best but also tough.