Can I deduct expenses for my sole proprietorship even if I claim the standard deduction?

Yes. Your expenses as a sole proprietor aren't deducted on Schedule A (Itemized Deductions). If you aren't using Schedule A, then you claim the standard deduction. Income and expenses from a sole proprietorship are entered on Schedule C (Profit or Loss from Business) or Schedule C-EZ (Net Profit from Business). Your net profit/(loss) is then entered on line 12 of Form 1040.