The easiest approach is a clear process you communicate in advance. Ask attendees to note their mileage or public transport costs and submit them through the expense system you normally use.
State the rules in the invitation already. Indicate the amount per kilometre that applies, whether public transport is reimbursed, and what the deadline for submission is. For large events, consider buying public transport tickets centrally. That saves administrative hassle.
A fixed allowance per region is an alternative: simple, fair, no individual expense claims.