In your second photo I can see that the plant has square stems with opposed leaves, alternating back and forth between the two sets of opposite sides of the stem. So....it is something in the mint family, though not necessarily in the Mentha genus. My best guess from the photo is that it is either Glechoma hederacea, better known as Creeping Charlie or Ground Ivy ,or it is Lamium purpureum (Purple Dead Nettle). If not, then it is likely a mint. In any case, little bits of stem or root left in the ground will produce more.
Fragrance would help in its identification. If it is one of the mints it will have a characteristic smell. Ground Ivy has a “sort of” minty smell. If it has little or no minty fragrance it is probably Purple Dead Nettle which, now that I think of it, is a likely suspect as it is vigorous locally this time of the spring, rushing to bloom in April. Let a piece grow in a pot until it blooms. Both Ground Ivy and Purple Dead Nettle will bloom in a few weeks. Purple Dead Nettle will have a terminal flower spike, and Ground Ivy will have individual booms in the leaf axils.
Despite reassurances from Monsanto, I would be hesitant to use Round Up where I would be planting food crops immediately afterward. Try pulling and digging as much as is practical now. Plant the tomatoes and peppers and keep pulling new sprouts. In late summer/early fall when production wanes, let the rascal flourish to get good growth and nail it with your Round Up when the plants are storing to the roots for the winter.
