
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.
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.