I have sole custody. Can I give my sister POA to act on my sole custody agreement to care for my child in my absence of 3 months?

If the father consents in writing to your plan you may allow your sister to care for your child. Dad is superior in the right to spend time with his child over your sister. Texas has a procedure and form to do what you want to do, but both parents must be on board. The law also requires you to obtain the consent of the court that issued the order of custody. In addition you may risk custody if you allow your daughter to live with anyone other than you for six or more months. 

The specific answer to your question may depend on what your custody order specifically says and by what you mean when you say "sole custody."     If you are named sole managing conservator (SMC) in the order then you are likely OK with letting your sister care for your child.  Be aware that if your medical processing is delayed you may have a problem.   One basis for someone to file suit to obtain custody is voluntary relinquishment of care, custody and control for at least 6 months.    If your sister has your daughter for more than 6 months, your ex could file suit to attempt to obtain custody.   How successful he might be depends on the reasons you were named SMC in the decree and other specific facts in your case.