Is it permissible to perform Umrah at another's expense, and is it considered an Umrah for someone who does not possess the financial means?

It is permissible for you to go for 'Umrah at your sister's expense, and your 'Umrah will be valid and sufficient if you perform its pillars and duties. Your sister will be rewarded for her assistance to you in obeying Allah. Performing the obligation of Hajj or 'Umrah is not affected by whether the person spent nothing or spent little on it, as long as the conditions, pillars, and duties are fully met.