Please note that this is a work around

Multipicklist fields unfortunately don't work in sharing rules. Here's a simple workaround:

Step 1: create a separate field, e.g. a checkbox which will reflect whether a contact is a vendor or not (you don't need to add this field to a page layout)

Step 2: create a simple workflow

- it should run "every time a record is created edited or did not previously meet the criteria"
- the criteria should be a formula something like INCLUDES(yourmultipicklist__c, "Vendor")
- add a field update to this workflow to set the checkbox to true

Step 3: use the checkbox in your sharing rule