This is probably going to be a bit of a weird one but here it goes.
I was looking to see what the easiest method would be to photograph the heavens during the day time in such a way that the stars are visible. Are there particular wavelengths or specific kinds of cameras that would be well suited for this task? Perferably on the more inexpensive side.
tl;dr: While #JamesK says you can't, I'm pretty confident you can do it with a camera lens or telescope with a large aperture (10 to 15 cm) as long as it's pointed in the correct direction and moving to track the star during your exposure.
Stars will be unresolved and so their apparent size will be determined by diffraction and astronomical seeing.
In arc seconds those are about 0.024 $\lambda \text{ (nm)} / d \text{ (cm)}$, and 1, respectively. Visually at 550 nm that works out to about a 6 inch (15 cm) aperture for them to be equal.
Daytime sky brightness of one square arcsecond is estimated to be about +3 magnitude, so if you can resolve that with telescope aperture of 15 centimeters then the very brightest stars and planets should be visible if your telescope is already pointing right at it.
It is going to be exceedingly difficult if not impossible to do it with your eye in daylight. Even if you put in eyedrops to dilate your pupil to 0.6 cm and use some kind of skinny tube and a black cloth to wrap your head to view only a tiny bit of sky, your eye will resolve an area of about 500 square arc seconds which now has a brightness of -3 magnitude.
In this case, in visible light, Venus is your only shot.
This is going to be some amount of work. There are various bands of atmospheric transparency, but your hardware needs to change to use them.
For near infrared (NIR) you have to get an image sensor where the IR filter has been removed (e.g. Raspberry Pi NoIR camera module) or remove it yourself. There are various tutorials, but note these are irreversible and somewhat destructive so do it on a throwaway camera.
You get modest improvement if you use a NIR camera and filter out the visible light. Rayleigh scattering that makes the sky blue scales as $\lambda^{-4}$ so if your average wavelength is now 900 nm instead of 550 nm the sky is roughly 7 times dimmer. Not a huge help, you'll still need a telescope with some NIR transmission, and of course at the wrong wavelength all the antireflection coatings on the optics become pro-reflection coatings so if you use a big camera lens you may have lens flare issues.
Silicon images sensors with all filters removed can have pretty good quantum efficiency all the way out to the band gap around 1100 nm. However absorption in the silicon charge collecting wells gets increasingly weaker so very thin sensors may not absorb all the light at the longest wavelengths.
On the other hand, there are thermal IR bands where the sky is fairly transparent and has a temperature of something like 200 Kelvin. If you can find an astronomical object that radiates strongly in the 5 to 15 micron band, you might have more than a snowball's chances in hell to image it.
Of course thermal IR lenses are made of pure, single crystal semiconductors like silicon or germanium rather than glass, so you will have to do quite a bit of work to put this together!
You can image Jupiter and Venus in daylight. You need to know where to point the telescope (as they won't be visible to the eye) so a "goto" system is useful.
You won't be able to image the day sky as the night sky at any easily accessible wavelength. Just putting a red filter won't work, light at all wavelengths is scattered, making the stars invisible at visible wavelengths. And the atmosphere absorbs infra-red, so this isn't an option.
If you do have a VLA you could do radio images. Not cheap(!)
