Is there something similar to the Greasemonkey script system for a browser in iOS?

Greasemonkey is explained in this Wikipedia entry.

I would like to add my own stylesheet to a particularly non-mobile-friendly website. Is there any way to accomplish this? Are there any browser apps that can do it?

You can use userscripts on iOS Safari using the new Shortcuts utility (though it seems to be complicated) and a lot easier using the free Safari Snippets app. However, this approach does not load userscripts automatically, they have to be manually loaded on each page load.

There seem to be at least two third-party browsers which support userscripts, the Gear Browser and the Alook Browser. I don't have experience with Gear, but in Alook you can create userscripts which are loaded automatically based on custom URL regexes (similar to Tampermonkey). While Alook has a one-time fee, note that Gear requires you to subscribe before you can use their userscript functionality (the monthly cost is currently higher than the one-time fee of Alook).

Finally, there is an external way of injecting userscripts into sites via a MITM-proxy approach. I found at least two repos, however haven't tried them out yet:

• https://github.com/SimonAlling/userscript-proxy
• https://github.com/eBay/userscript-proxy

Note that encrypted websites complicate the proxy approach as you need to decrypt your HTTPS traffic using a certificate.

At the moment there is the GreasePocket project in the works.

Have a look at their website, but it's far from a finished product, it will also probably need to be run on a JailBroken device.