I would start advertising for rent, put a for rent sign in the yard, and schedule some showings. Make sure you give them proper notice for entry for any showings. If they have an issue or have decided they want to stay, you'll hear from them really quickly.
Maybe stop by "after hours" and knocking on the door?
I know many leases give permission to the landlord to enter the property with adequate notice. So if you are wanting to make an inspection of the property, give them notice of that as well.
But it does sound like they intend to leave. So you should protect yourself and find a new tenant.
You are not obligated to rent to the current tenant if they have not indicated to you that they intend to stay. So, if you find a new tenant, you then can request the current tenant to leave according to your lease terms, and if necessary evict them according to the terms.
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