Yes, you can use multiple clauses in a sentence. The clauses need to be parallel in structure. Your clauses are okay grammatically (I am..., have been..., have.) but the sentence is a little clumsy (which I think you sense). With a long sentence, you will want to delete unnecessary words and make the clauses strictly parallel.
I would suggest rewording your sentence to something like this:
On the managerial side, I have extensive experience with accounting software, a history of working with this product throughout its life cycle*, and a demonstrated talent for improving existing products.
(*I'm not sure of the meaning of this clause in your original sentence, so I am not sure if this is what you meant. It might also be better to name the product where I have this product.)
In this revision, I use the verb have once and each clause is now a noun clause...experience, history, talent.