Is it possible to build a inverter that operates at a frequency set by an LC tank using ONLY that LC tank and transistors?
A LC tank circuit is just a inductor and capacitor connected to resonate at a particular frequency.  These are not active components.  They can not make a frequency that doesn't already exist.  They can filter, sometimes with quite high Q, a mix of frequencies to pick out or reject a narrow range.

In other words, A L-C tank can be used as the governing elements that set the frequency, and also to filter out spurious frequencies to leave you largely with a sine, but the original signal with the desired frequency content can only be made with something that consumes power.  This requires at least one active device somewhere.

If it were a desing criterion for some reason, it should be possible to create a circuit that oscillates and drives a L-C tank with a single active element.  It will be easier with multiple active elements or multiple windings around the same core.  If you are looking to have it produce significant power, I wouldn't worry about the number of active elements and think about drive level and efficiency instead.