The simple answer is no. One does not have to have the indwelling of the Spirit to believe. The New Birth accomplishes this need. The proof of this point is in the Scriptures and by asking simple questions to guide ones thoughts.
Were there not believers in the Old Testament economy? Of course there were. Did they have the indwelling of the Spirit? NO. However by requirement, they must have been born anew by the same Spirit. What requirement you may ask? The requirement revealed by our Lord at the end of the Old Testament economy and should have been known by the Old Testament teachers:
John 3:5 Jesus answered, Verily, verily, I say unto thee, Except one be born of water and the Spirit, he cannot enter into the kingdom of God.
John 3:6 That which is born of the flesh is flesh; and that which is born of the Spirit is spirit.
John 3:7 Marvel not that I said unto thee, Ye must be born anew.
Were the twelve disciples believers in Christ? Well eleven were and one was a fake. But yes, the eleven were believing on Christ. Did they posses the indwelling of the Spirit? NO However they had been born anew and had the abiding presents of the Spirit and later would have the indwelling Spirit in them.
What is the promise of our Lord as to the coming of the Spirit. Let's see what Scripture says.
John_14:16 And I will pray the Father, and he shall give you another Comforter, that he may be with you for ever,
John 14:17 even the Spirit of truth: whom the world cannot receive; for it beholds him not, neither is knowing him: ye know him; for he abides with you, and shall be in you.
1) He is to be the believers comforter. Christ was departing and this was traumatic on the believers being left behind.
2) Here we are told that the Holy Spirit is the Spirit of Truth - of course He is, no believer would deny this.
3) The world cannot receive this Spirit, why not? First, because the world here is the non-elect, they do not know Him. They do not know Him because they have not been born anew. Being born anew introduces the Spirit to the future believer and this self same Spirit abides with this new infant child while he takes his first Conversion steps. Like a father protecting his child from the evils of the world. This begins the baby steps towards Christ and the Gospel is the beacon which leads the way.
4) That in the near future the Spirit would be indwelling them.
What is the purpose of the Spirits indwelling? Again, what does the Scripture say?
John_14:26 But the Comforter, even the Holy Spirit, whom the Father will send in my name, he shall teach you all things, and bring to your remembrance all that I said unto you.
1) He will teach us. He does this by illuminating our minds as we study, witness and hear the word expounded. But He does not teach what we have not been exposed to. He does not give us information but aids in our understanding, of those things which have been revealed in Holy Writ.
2) He brings us into remembrance. As humans, without this aid, we would forget more than we know.
John_15:26 But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall bear witness of me:
1) The Holy Spirit does not seek His own glory but moves all things towards Christ in our knowledge, so that Christ is further exalted.
John_16:7 Nevertheless I tell you the truth: It is expedient for you that I go away; for if I go not away, the Comforter will not come unto you; but if I go, I will send him unto you.
The Holy Spirit was so necessary for the Universal Church, which Christ is building, that the Lord Himself said - it is expedient for you that I go away. Why so expedient? The answer I believe is in this verse:
1Pe_2:9 But ye are an elect race, a royal priesthood, a holy nation, a people for God's own possession, that ye may show forth the excellencies of him who called you out of darkness into his marvelous light:
Other passages, tell us more about the Spirits office in the Universal Church.