dimkasmir,
The short answer to your question is no. The coil from Fry's will not work.
The Fry's coil has far too little inductance, and lacks a primary winding.
Unfortunately, the Tubes and More website has just changed; last night the specified coil could be found, today it cannot! ( And where did the Google Cache feature go? Google still has links to the old pages that are now gone. )
I did look at the information on the specified (AES P-C70-RF) coil last night. This appears to be sold for use as an RF interstage transformer for the MW AM broadcast band. It is made rather like the classic "vari-loopstick", but the slug is probably threaded and screwed directly into the coil form, without the threaded brass rod sticking out the metal mounting clip at the end. There is a separate primary winding on this coil. There was no inductance specified, but as a replacement part, it would presumably work with the typical range of variable tuning cap values of about 140 to 400pF, maximum.
It is possible to determine some of the necessary specifications for the coil by analyzing the transmitter schematic.
On the schematic, the coil winding terminated at 1 and 2 is part of an LC resonant circuit. This circuit determines the transmitting frequency. The coil winding terminated at 3 and 4 is the primary. It couples energy from the tube's plate into the resonant circuit to make the circuit oscillate at the transmitting frequency.
Most of the capacitance for the resonant circuit is 150PF C6. There will be additional capacitance due to the tube and stray capacitance. Perhaps 15pF more, as a guess. This gives us 165pF as the resonating capacitance.
If we assume a 1MHz resonant frequency, about the center of the MW AM broadcast band, we can calculate the necessary inductance to resonate with the 165pF capacitance. This gives us 154uH as the nominal value for our coil. You will want an inductance value reasonably close to this.
So the Fry's coil you asked-about, at only .151 (uH ) to .209 ( uH ), comes-up a bit short. I looked for a suitable coil at Fry's and found nothing. ( Fry's used to be an electronics shop. Now a search for "transformer" turns up mostly plastic children's toys…)
The AES part may still be available, you might check with them. Other vendors of parts for old radios sell similar coils.
There are many possible alternatives, depending on what you want to do. You might buy a new coil, salvage one from an old radio, or wind your own from scratch.
If you want a coil that looks like the one at Fry's, you might try one for ( or from ) a transistor radio. Mouser still sells 455kHz IF transformers that might be adapted. The associated oscillator coils are gone from the Mouser catalog, but would also work.
Tube-type MW radio oscillator coils or IF transformers might be a better option. They would be larger, and easier to work-with.
Some choices for the coil would require some modification of the transmitter circuit to adapt it to the particular coil. Coils with low voltage insulation might be safer with an isolating capacitor on the primary. Or, you might change the circuit to a Hartley oscillator, which can also work with a tapped coil, rather than one with an isolated primary winding.
The value of C6 may have to be adjusted to get the frequency you want with a different coil. You might want to substitute a variable capacitor for C6. This would be particularly useful if the coil you use is not adjustable.
You did not say where you are located; if we know where you are, someone may have better suggestions about parts suppliers.
An old junker AM radio might be your best source for parts. It would have the coil and a tuning capacitor.
If you want to wind your own coil, look for articles on building crystal sets and regenerative receivers; the same coils should do the job in your transmitter.
Be sure to come back here with more questions as they arise.
Good luck finding all of your parts.
Ted