What could go wrong? Well, the tank blowing up, for one thing.
Actually, there are procedures for hot-tapping and similar operations, discussed to some extent in API-653. If the welding is far enough below the liquid level that it can't ignite vapor, and is minor enough that it's not weakening the structure, then it can be done.
It would be best to have someone more local that was familiar with this type of operation and the tank in question to go through it to determine a suitable and safe procedure, if feasible, or if not, to justify draining and cleaning the tank.