Congressional Republicans have found a way to use the federal budget to open the Arctic National Wildlife Refuge to oil drilling—and they don’t plan on stopping there.

GOP leaders in the House and Senate are exploring ways to also expand drilling in the Arctic and Atlantic Oceans through congressional budget rules that allow them to pass major policy changes on a simple majority vote.

“There are other opportunities for us,” Sen. Lisa Murkowski (R-Alaska), the chairman of the Senate Energy...