The other two machines have their codes in the Forest. There's 3 codes in there:
1. On a plaque in the first Forest area (head directly to the left and up)
2. In the Forest Cave (check the bush in the middle of the room)
3. In the Tent with the Blue Slime girl (check the papers on the table)
4. The MDP code is found in the trash piles in the Lab.
But yeah, the vines are access to the Slime Village, which won't leave until you recruit the Slime Girl from the tent. If you ate her, there's no way passed them.