Is it possible to use a YIG tuned filter to make a YIG tuned oscillator?
YIG tuned filters use several spheres for a high order filter.

Unless you take the filter apart and assemble the amplifier on the coupling to the spheres, the electrical length in and out of the filter will dominate the phase shift round a sustaining amplifier, meaning you will probably need additional switchable phase shift to be able to oscillate at all frequencies.

There's a lot of knowhow, aka technology, goes into the making of a YIG oscillator. It's difficult, and the people that do it for a living know more about it than you, or me. Cobbling together an oscillator out of a huge filter is rather like trying to build a cheap 4-wheeled vehicle by welding two motorbikes together. By the time you've solved the essential problems, you'll have taken a lot of time to be left with something far inferior to buying a Fiat 500.

You can get very wideband varactor tuned LC oscillators. You should be able to cover the 1 to 10GHz band with 3 or 4, for cheaper and smaller than a YTO, if you really have a generous phase noise allowance.

Depending on what you mean by 'continuous' - no missing frequencies, or the ability to sweep from one end to the other with absolutely no dropouts - you could use a 2-10GHz YTO and a divide by 2 IC to cover the bottom octave. Or maybe a 5-10GHz YTO (which ought to be smaller and cheaper) and a divide by several to cover the 1 to 5GHz range.