So I used sp_blitz and came across a suggestion to change verification to checksum. What I want to know is, would this step require any downtime?
Nope, no downtime for this change at all.
Here's the explanation of what/when things happen:
Keep in mind that just changing the setting to CHECKSUM does not instantly add CHECKSUMs to your existing data pages in the database. In order for this to happen, you have to read each page into memory, make some sort of change and then write it back out to the storage subsystem. This can happen from normal INSERT/UPDATE/DELETE activity over time, or from rebuilding your indexes.