Without knowing all the dynamics of your position and the culture, my general thinking would be as follows:
1. Wait until after probation is completed since it's so near anyhow.
2. After successfully passing the probationary period, come up with an actual figure of what it is costing you to drive around. The IRS uses a figure of about $0.57/mile if that helps.
3. Determine if it's worth it to bring it up compared to your salary and benefits. And this will depend on the culture of the company, your boss, past precedent (if any), and lots of other factors. For example (using big numbers for added effect), if you earn $200k/year and it's costing you $200/month to drive around, and you expect your boss would view bringing it up as a cheapskate/ungrateful mindset, then I'd leave it alone as the cost of doing business. On the other hand, if driving around is eating up 10% of your take-home pay it probably makes sense to bring it up.
4. If you do bring it up, frame it with actual numbers so your employer understands the real impact instead of vague ideas of "I drive around a lot," and ask for suggestions rather than presupposing a solution. For example, besides working from home, solutions could include reimbursing you directly for costs, providing you with a company car, etc.
5. Finally, you might bring up working from home even if the costs of driving around aren't high, but be prepared to explain how you can still easily accomplish the mission without driving to these sites. Be honest with yourself here to make sure you really can do it.