You can build the logic to calculate and display your shipping date using custom code; however-
Timezones. Depending on where your user is, you need to account for their local time difference. Server-side also overcomes this however you’ll want to make you’re language clear e.g. “Orders before 11am PST will be shipped on the next shipping day ( click for details ).”
Cutoff time. If they start the order at 10:58 am, they’ll see a different shipping date. I think clear langage is the best way to mediate that confusion.
If you don’t have any devs, feel free to PM me and I’ll give you a quote.