Yes, pure silicon can form sp3 hybrid orbitals.
"The 3s and 3p orbitals of Si are filled by 4 electrons. A single atom has [Ne] 3s2 3p2. But when multiple atoms get together they do so in order to minimize the overall energy. And to minimize the overall energy, the 3s and 3p orbitals hybridize to form 4 tetrahedral sp3 orbitals. And the Si atoms get together by joining their 4 sp3 orbitals. Each sp3 orbital has one electron and is capable of forming a bond with a neighboring atom[, and] each atom has 4 neighboring atoms." [Reference https://www.physicsforums.com/threads/silicon-bonding.742932/]
I hope that helps!