Employment Law Questions? Ask an Employment Lawyer.
Is there an agreement or policy within their handbook that states they reimburse expenses?
The written policy is somewhat indefinite - "they will make a timely effort to reimburse". But, you have a long history of the reimbursing and others were reimbursed. But the bigger problem is that you'd have to sue them in court to get your money because your state does not mediate or investigate reimbursements on employees' behaves. So you could claim an implied contract for reimbursement and you'd win.
Now, in regard to the non-compete, these are separate issues. The non-compete may or may not be binding on you. Most non-competes in existence aren't enforceable because they do not protect a legitimate business interest, which courts require to enforce a non-compete. There is even a sandwich chain that requires sandwich makers to sign non-competes - those non-competes certainly aren't enforceable, but what is someone who makes near minimum wage going to do to invalidate the agreement in court? If you want to tell the employer that you'll compete because they breached your employment contract by not paying the reimbursements...that way you avoid the cost of having to litigate to have the non-compete ruled invalid. Or you could just go and compete and see what if anything they do - if they challenge you, then counter-sue for the reimbursements.