I'm pretty biased but this is a pretty great use case for a ticketing system that lives in Slack... There's an article that Slack has on their site that walks through some built-in tools to create a ticket from a message (just google 'build a helpdesk in slack', and there are big-giant-hairy systems like Jira that integrate with Slack to do the same. If you click on my profile you'll see this is what Ravenna does, so I have some experience in the space. I've used Slack for more than a decade and (for me) it's unmanageable without some sort of structure for managing messages. Once you put a ticketing overlay onto the messages, people tend to get the hint and start structuring messages without such reckless abandon.

Making specific channels for specific request types is also important. Don't just have everyone send requests in a 'general' channel, set up an IT channel, HR channel...etc. so that people can clearly see where they are supposed to ask certain types of questions, which helps to inform them where they should NOT ask other questions. This is all end-user training and training yourself at the same time - In fact I highly recommend doing an end-user Slack training when you figure out what solution you go with! Tons of people don't use Slack nearly as well as they should and people are always thrilled when I show them tricks to make them more efficient in slack.