@Joseph Senno I would think most improvements would be 100% on the landlord unless the tenant wants to select colors, etc, or it is being done to repair tenant caused damages.
I am assuming the tenant has been there a while and is a good tenant. I don't have advice but I personally might replace the floor and take a certain amount of money as a refundable pet deposit that you will give back on move out if the floor is in good shape. That way nothing is "owed" the tenant...if he partially pays for the floor he will rightfully feel he has some sort of equity in the floor. That you should keep rental rates steady for a term or something? I rented an apartment for a year and I paid up $10 month for my floor plan supposedly because it had an extra foot overhang of countertop in the kitchen. You could also amortize the floor cost over 8-10 years and raise rent that amount.
And yes, floating solid core vinyl planks right over the existing sheet vinyl is a great solution. It will be beautiful, easier to keep clean than damaged flooring and should last a decade or two even with pets. Keep your tenant happy because you will have to replace the floor anyway if he vacates and you need to relet.