If you covered it by reducing their rent you don’t have to do anything since your reduction in rent income balances your expense. If you paid them by check, that is your proof of that expense. 

So your tenant paid a utility bill and you agree to reimburse the tenant?

I would simply give them a discount on their next month's rent.  You would not report the expense as an expense you paid, you just report the lower amount of rent as income.  That has the same effect on your taxes and is simpler.

(In other words, suppose the rent was $1000 per month and the utility bill was $300.  Whether you report $12,000 of rental income and a $300 expense, or simply report $11,700 of rental income, your tax position is the same.)

Would an email transfer be ok as a receipt?

A receipt for what? If you give them a rent reduction, you simply book the rent that you actually received.

If you intend to book the full rent and issue a separate check for the expense, and deduct the expense separately, here is your problem:

You can only deduct ordinary and necessary expenses of renting the property.  The expense must be billed to you, or if not, you must have adequate proof that it was an expense of renting the property and that you were ultimately responsible for it.  Your canceled check to the tenant is proof that you paid the tenant, but it is not proof that you paid a legitimate expense connected with owning and renting the property.  You would also need a copy of the utility bill and any other records that would tend to show that even though the tenant was billed for the expense, it was ultimately your responsibility as property owner.

That is why I say it would be far simpler to reduce the tenant’s rent for one month or longer, however much is needed to make the tenant whole for this expense.