Hi, welcome to Zendesk.
Quick honest answer up front: there isn't a native way to restrict ticket forms by user type. It's a known gap and there's a product request with a fair bit of community support behind it (search "Use User Segments to restrict access to Ticket Forms" in the community ideas section if you want to upvote it), but right now the platform just doesn't have a built-in toggle for "only show Form X to Students."
That said, there are a few real-world patterns that work, depending on how clean you need the separation to be.
Option 1: Two brands, two help centers. This is probably the cleanest fit for education. You set up one brand for Staff/Faculty (e.g. helpdesk.yourcollege.edu) and one for Students (e.g. studenthelp.yourcollege.edu). Each brand has its own help center, its own set of ticket forms, its own visual identity. Behind the scenes it's all one Zendesk instance, same agents, same workflows, but the front door is fully separated. Tickets from each brand can be tagged automatically so your agents can still tell them apart in views. This is what a lot of universities and colleges end up doing because the staff and student experiences genuinely are different and benefit from different branding anyway.
Option 2: Theme JavaScript to hide forms based on user tags. You tag users as "staff" or "student" (ideally automatically via SSO if you've got that hooked up), then add JavaScript to your help center theme that hides ticket forms a user shouldn't see. This is the workaround Zendesk effectively acknowledges in the product request thread. It works, but it's fragile: theme updates can break it, it's client-side so technically bypassable by anyone determined enough, and you need someone comfortable with the help center code to maintain it.
Option 3: One form with conditional fields. First field is "Who are you? Staff/Faculty or Student?" and the rest of the fields appear conditionally based on that. Doesn't actually separate the forms, but functionally splits the experience. Cheapest to set up and maintain. Downside is anyone can pick the wrong option, and if your staff and student workflows diverge much (different SLAs, different routing, different required info), one form gets ugly fast.
For a community college I'd probably lean toward Option 1 unless your IT team is small and the visual separation isn't that important. The two-brand setup also gives you room to grow: separate macros, separate triggers per brand, easier reporting on staff vs student volumes, etc.
One thing worth noting: even with Option 1, you'll want to think about SSO. If students authenticate one way (e.g. student SIS) and staff another (e.g. staff AD), routing them to the correct brand's help center via SSO redirect rules is what makes the separation actually stick. Without that, anyone can just type either URL.
Happy to go deeper on any of these if it'd help.