Jontz
If the Philco dealer left the "J" bolts in place when the set was sold to the original owner, and then assuming subsequent repairmen also left them in place, then you might get away with transporting the set intact.
But I wouldn't do it, even if the chassis is still bolted in with those special "J" bolts - and chances are that it isn't. The "J" bolts were supposed to be removed when the dealer delivered the new 37-116 to the original owner, with the chassis left to "float" on the original rubber washers which are likely also gone with the wind.
In short - do remove the chassis, and the 14 inch speaker. Your back will thank you.
IMPORTANT NOTE
Be very careful when removing the large speaker! There is a conical piece of wood in front of the cone. One wrong move and you'll tear up your speaker's cone! I suggest removing the chassis first - remembering to unplug the speaker wiring harness from the back of the speaker - and then laying the cabinet face down on a thick blanket. Remove the two bolts and nuts holding the speaker in place on a wooden support, and then
carefully pull the speaker straight out (NOT "up" as in "up toward the top of the cabinet" but OUT (as in "straight back"), as much as possible as the wooden support which holds the speaker in place will interfere somewhat.
One more thing. Early 37-116 models have five individual push pins that connect at the audio output transformer, mounted on the back of the speaker. You'll have to remove a cover where the wires go into the audio output transformer, make notes of which wire goes where, and then just pull each one out. Later models use a five-contact socket and plug; the socket is on the speaker wiring harness, and the plug is mounted on the back of the speaker. Check for the presence of a screw which may be holding the socket to the plug and remove if necessary. You'll recognize this by the telltale screw head at the back of the socket where the wires come out and go to the radio chassis.
Be careful, be safe, follow the suggestions given by Codefox, Steve, and myself - and enjoy your "new" find!