Can I replace an extension cord plug myself?

Yes, but do it carefully. Cut off the damaged plug, buy a heavy-duty replacement plug that matches your cord gauge (e.g., 14 AWG or 12 AWG), strip the wires, and connect them to the correct terminals: brass for hot (black), silver for neutral (white), green for ground (bare copper). Disconnect the cord from power first. Don’t use a lamp-cord plug on a heavy-duty cord, and toss the whole cord if it’s frayed or cracked.