Auto-mailing a check to a non-member

We have a member who we've agreed to cover rent for them. Their landlord is not a member.
This will be a regular occurrence for the foreseeable future.
Currently we have to physically print the check and mail it, every month.

Can this be automated online somehow? e.g. direct deposit? If that is not an option, can the church dispatch a check and mail it to them? (and can that process be triggered online?)
What are my options here?

If you are in the United States, a check to a landlord, whether they are a member or not, is the only way you can pay them after the bishop or branch president authorizes payment with fast offering funds. In other countries different rules may apply, but I doubt it can be automated.

Although you correctly identify the inconvenience involved, trust me that this is very worthwhile. Having each individual rent payment be manually processed helps avoid those situations where funds end up not being correctly used. At the start things seem clear, but I've been surprised at how things can change later on without leaders realizing all the impacts. A member gets a job but the ward still pays rent, the member moves but the ward still pays rent, a friend moves in but the ward still pays the full rent, etc. Leaders being required to manually process these type of fast offering expenses is a helpful way for the Bishop to remain engaged, and also be a proper steward of the sacred funds.

Your specific situation may never fall into any of these categories, that is usually the case. But I've been grateful in the past on occasions where this process has helped to not let things slip past me.