By "constantly accessed" do you mean "I don't have any downtime or maintenance window in which to make critical changes to the database"? If so, then that is the scary part. Like everyone says, this update will require an exclusive lock the table, and if it has to wait for that lock, then you may get blocking, timeouts, and possible irate users.
Too, the new column may require updating the table. I'm not 100% clear on what happens here, but if (this is a "what if" example, but it could happen) you add a char(100) column to a 10,000-row table that had just had its clustered index rebuilt with fillfactor = 0, then you are adding 100 bytes per row to pages that are already full, and where does that data go? Either you get page splits or forwarded records, both of which are going to take time for SQL to build -- meaning long blocks and further delays to access requests to the table.
Seriously, try to find downtime, or at least low-volume time (Sunday 2am?) to do work like this. And test it if you can... but very few places have testing sites that emulate seriously busy Production websites. If nothing else, a test on a copy of you DB will show roughly how long it will take, if you can get a downtime window.