Technically, you *could* passthrough vape on any mod that has a USB port. However, there's a risk of overloading your USB port. So yes, it does depend on your mod. Some mods do have passthrough support. I did a bit of research on the subject, and here's what I found out:
You will need to know how much power your charger is putting out and how much power your USB port can handle. Your USB port is only equipped to handle so much current from a charger while vaping. (Current is basically a measurement of how much electricity is running through your circuit, or in this case, running through the charging cable.) If the battery tries to pull more power than the port can handle, it can do some serious damage to the port. I mostly learned that part the hard way while trying to passthrough on my Hohm Slice. The USB port on my Hohm Slice is useless now, so please be careful so you don't make the same mistake I made.
Here's a bit of information on passthrough vaping:
http://blog.misticecigs.com/what-is-pass-through-vaping
And as for your question about which cheap mods support passthrough, you could try the Innokin Coolfire IV if you prefer not to go too high on your wattage. That one will go up to 40w. Or if you want a bit more power, the Eleaf iStick supports up to 100w and supports passthrough. That one's pretty comparable to my Hohm Slice as far as quality, and my Hohm Slice delivers a great vape for me without going too high on power. Use Google to find those.