Yes. „Vă rugăm” would mean that you and at least another person are asking someone to do something. „Vă rog” means that only you are asking. Literal translations should help you understand better:
Vă rog - I ask/am asking you (to do something) - formal (or informal when talking to more people)
Vă rugăm - We ask/are asking you - formal (or informal when talking to more people)
Te rog - I ask/am asking you - informal (just one person you are talking to)
Te rugăm - We ask/are asking you - informal (just one person you are talking to)
This actually works with every pronoun (except first person, obviously): o rog/o rugăm (asking her), îl rog/îl rugăm (asking him), îi rog/îi rugăm/le rog/le rugăm (asking them).
This reminds me of Polish, where there are two different versions of "thank you", depending on whether somebody is alone - "dziękuję", lit "I thank you", or in company - "dziękujemy", lit. "we thank you" (and I always forget to use the plural form). I was amused the other day when opening the gate for a neighbour and her dog to hear "dziękujemy".