
Can anyone order someone
else's Texas birth certificate?

No.
If you’re a Texas resident you can only order your own Texas birth certificate; or the birth records of immediate family members.
See Section 181.1(13) of the Texas Administrative Code for what qualifies as an immediate family member.
https://texreg.sos.state.tx.us/public/readtac$ext.TacPage?sl=R&app=9&p_dir=&p_rloc=&p_tloc=&p_ploc=&pg=1&p_tac=&ti=25&pt=1&ch=181&rl=1
For non-Texas residents, you can only order your own Texas birth certificate, or the birth records of your own children if you’re listed as a parent on the official birth record.