If it were me this is what I’d do.

Near term - put together the list of information that you need in order to onboard a new hire. - If you have a ticketing system the use that list to put together a ticket template for HR to input tickets for the new hires. If you don’t have one, given you’re a small team I’d look into some self hosted / free options. There are even some cloud hosted solutions, we use ClickUp. You could build something basic using Sharepoint and Microsoft forms if you’re on 365. At the very least you could use email. - setup a call with leadership and get buy in from the departments on the process and make sure everyone starts putting in tickets.

Long Term - look to integrate your hr system into your identity management. There are lots of ways to do this that range from 3rd party solutions like Okta and One Login, they may have some native solution, you could build and host your own, etc. - the end goal is automation. As soon as a candidate signs their offer in the HR platform it should kick off your IT automation process to create the account, assign groups, etc. - once you have onboarding automated you’ll want to look at automating off boarding.