Access to opponent's computers

If I were involved in a lawsuit in Texas, could I get access to my opponent’s computer’s hard drive?

Maybe. As a threshold to granting access to electronic devices, you (the requesting party) would have to show that the responding party somehow defaulted on its obligation to search its records and produce the data you requested. See In re Shipman, 540 S.W.3d 562, 568-69 (Tex. 2018) citing In re Weekley Homes, L.P., 295 S.W.3d 309, 317 (Tex.2009)(orig. proceeding). Courts do not rely on skepticism or bare allegation that a responding party failed to comply with its discovery duties to grant access to electronic devices, but rather evidence is needed. See In re Shipman, 540 S.W.3d at  568-69.