You will find this to be an almost impossible DIY task, for a long list of reasons.
Tungsten wire is difficult to form into shape and very difficult to manipulate in thin gauges. Specialized tooling will be required to coil the filament.
The electron emissivity of the wire will determine the physical dimensions of the gun assembly, which must be determined experimentally because the as-manufactured state of the wire will affect this and cannot be predicted in advance.
It is possible to use an indirectly heated electron emitter, in which the filament heats a sleeve it is inserted into which has been coated with an alkali metal oxide mixture that has very high electron emissivity. the sleeve is then part of the overall circuit for the gun, separate from the filament, which will allow you to optimize the designs of the heater and emitter separately. This is common practice in high-vacuum electron tubes but it requires detailed knowledge of how to compound and apply the coating.
To route the electrical connections out through the glass wall of the gun tube requires the use of a special metal alloy which has been engineered to possess precisely the same thermal expansion coefficient as the glass it is stuck through. You'll need to find a source for this (highly specialized) material.
Once the tube has been assembled and evacuated it will bake out at high temperatures, causing all the components of the gun assembly to outgas and thereby ruin the vacuum inside the tube. To deal with this you need to include a component called a getter inside the tube which will adsorb or chemically react with the outgassing and a getter flash which will react with any residual oxygen that either leaks into the tube when it is hot or outgasses from the glass envelope of the tube during operation.
If you are sufficiently clever you can part the entire gun assembly off the neck of an old-school cathode ray display (TV) tube after carefully breaking the vacuum in it and then find a glass-blower who can weld it into the device you want to use it in, but you'll need to restore the getters because they will be wrecked upon exposure to ambient air.
