I realize that animals don't have souls so that they don't automatically go to heaven but do you think that God would allow us our pets in Heaven if we were to ask him? I've sort of promised my dog that I would make sure he could come be with me there forever.
I can't speak for God where he has not spoken. However, if animals don't have souls, it appears there would be nothing of a pet that could go to heaven. If they do have souls, although the Bible is pretty much silent on the matter, then it would seem possible to ask God to allow you a pet in heaven. The Bible really doesn't say one way or the other. It doesn't appear that it would be wrong to pray to God that if he allows animals into heaven that he would allow your dog to accompany you.
On the other hand, your question is somewhat related to the old question that says, "If we recognize people in heaven and my parents/wife/child/pet is not there, won't I be sad? Yet we know that there is no sadness in heaven." I suspect, although there is, again, no scripture to support this directly, that being in the presence of God will be such overpowering joy to one who has given his life over to trying to be like God that one will not miss those who are not there. Another suggestion is that we will be so filled with knowledge of God and righteousness that we will understand and agree with God as to why our loved ones or pets are not there.
I can not properly answer your question from the Bible, because the Bible doesn't address that question. What it does address is whether you will be in heaven or not. Regardless of your love for your pet, the main thing is to make sure you will be there.
Note: an observant reader pointed out that the Hebrew word for soul is used for both animals and man. Thus animals are souls. However, the New Testament does make a distinction between the soul (life-force of man and animals) and the spirit which lives on after a man dies. It is this spirit that the Bible is silent about in relation to animals. So in the first paragraph of my answer, substitute the word "spirit" for "soul" and the answer is still essentially valid. The Bible is silent as to whether there is anything in an animal akin to what of man will be in heaven.