The University offers laptops to every full-time student living on campus. The laptop is free, and if you remain a campus resident all four years, it's yours to keep. However, unless you are a student in the Honors College, you will receive a netbook as opposed to a full-fledged laptop. The netbooks are still pretty nice, and they're easy to carry around, but remember that a netbook is generally meant to be a smaller, cheaper, and less powerful companion to a laptop or a desktop PC. Therefore, it may be wise to bring or buy your own computer if you plan on using it for more than just doing schoolwork or surfing the Web.
You will have plenty of access to wireless Internet at SLU. The wireless network does tend to run slightly slower at peak hours, such as mid-day during the week when everyone is in class or rushing to get papers finished before class, but it's never painfully slow. There are plenty of computers around campus. The Learning Resource Center and the library have computer labs, and overcrowding usually isn't an issue. The only time it may be hard to find a spot there is during mid-terms and finals week.