Punctuating the sentence
The sentence as you punctuate it has three commas—two marking the ends of introductory phrases, and one separating the sentence's two independent clauses. In my view the third comma occurs at the place most in need of internal punctuation. Because it performs a more basic demarcation task than the other two, I would be inclined to emphasize its preeminence either by keeping the other two commas and replacing it with a semicolon:
Since you are a Salaried Exempt, you technically don’t need to submit time; but with Ernst & Young, we want to track your hours.
or by keeping it as a comma and deleting the other two commas:
Since you are a Salaried Exempt you technically don’t need to submit time, but with Ernst & Young we want to track your hours.
I see a parallelism of form between "Since you are a Salaried Exempt, you technically don’t need to submit time" and "with Ernst & Young, we want to track your hours" that provides a rationale for treating both introductory phrases similarly—either with commas at the end of both or with no punctuation after both. But the motive for making this choice is as much for aesthetic symmetry as to emphasize to the reader that the sentence consists of two syntactically similar halves (intro–assertion and intro–assertion) bound by a conjunction ("but").
Admittedly, my preference here either for one comma or for two commas and a semicolon may be purely idiosyncratic. I have noticed that many people are far less disturbed than I am when (as in this case) commas operate at different hierarchical levels in a single sentence.
Capitalizing 'Salaried Exempt'
A case could be made in favor of capitalizing a job title, such as vice president of human resources:
Since you are Vice President of Human Resources...
although style guides by no means universally approve of this practice.
But adding initial caps to employment status—such as "salaried exempt," "salaried nonexempt," "full-time hourly," "part-time hourly," or "unemployed"—receives essentially no support in the style guides I am familiar with. So I would leave that status in your sentence lowercase.
I would also remove the indefinite article from before the status. In effect, "you are salaried exempt" is a short-form way of saying "your employment status is salaried exempt." To my mind, the most natural and intelligible way to express that idea is to say
Since your employment status is salaried exempt, you technically don’t need to submit a time sheet; but with [or at] Ernst & Young, we want to track your hours.