The Repair Clinic website has a whole page dedicated to what could cause your washer to shake or vibrate.
[link|https://www.repairclinic.com/RepairHelp/How-To-Fix-A-Washing-Machine/42-6-150335-/New/Kenmore-Washer-Vibrating-Or-Shaking-16612690|Kenmore Washer Vibrating Or Shaking - Solutions & Repair Parts - Repair Clinic]
Based on their suggestions, I'd try rotating the inner tub by hand to see if it's related to a bad transmission or a damaged inner tub. Apparently your washer has something called a balance ring that's supposed to be filled with fluid to balance the tub, so you're supposed to verify that it still contains fluid.
Those are the main things that stuck out for me, but you can certainly look through them yourself and see if anything else may be a possibility.