A shared inbox can handle support until the first customer asks what response time they are actually promised. At that point you need tickets, owners, and clocks.
Tickets that know their customer
Each ticket belongs to a company, and can be linked to a contact and to the contract that covers them. Tickets carry a status (Open, In Progress, Waiting on Customer, Resolved, Closed), a priority from Low to Urgent, and an assigned owner on your team.
Deadlines from the contract
Contracts can carry first-response and resolution targets in hours. A ticket takes its deadlines from the customer's contract, and falls back to defaults when the contract does not specify any. Both clocks are set when the ticket opens.
A warning before a deadline passes
A background check looks at open tickets and creates a reminder for your team when a first-response or resolution deadline is close or has passed. Each clock alerts once, so one ticket does not nag you repeatedly.
A narrow view for customers
Business clients can sign in to a portal to see their own tickets and open a new one. They cannot set priority, assignment, or contract, because a client should not decide their own deadline or who handles it. Tickets you have moved along since they last looked are marked as updated.
Attachments and history
Files can be attached to a ticket, so screenshots and logs stay with the issue rather than in someone's downloads folder.
Starting small
You do not need a service desk to benefit. Even one list of open tickets with owners and deadlines answers the question every support conversation eventually asks: what is waiting, and for how long.