The sensor in digital camera are sensitive to IR but many have IR blocking filers built in. These filters may be a separate optical part or more likely in low cost cameras is a coating on the lens. Some do not have an IR blocking filter.
Try to find the spectral response of the camera from it data sheet to see what its response to IR is.
Typically only B&W cameras respond to IR. Color cameras have RGB filter over the pixels to only pass that color (wavelength). However the Red pixels could have a good response to IR. Again typically there is an IR blocking filter.
The older Web cams had a separate IR filter that could be removed. Newer ones tend to have the IR blocking coating on the lens. Again look at the spectral response of the camera from it data sheet to see what its response to IR is from any camera you are considering.
A source of info about various Web Cams and low cost cameras can be found by searching for Web Cams used in Astronomy. Here is one site:http://homepage.ntlworld.com/molyned/web-cameras.htm
Here is a tutorial showing how to mod a web cam for IR.http://www.instructables.com/id/Infrared-IR-Webcam/
Do more google searches as there is a good amount of info out there on this subject.
Lastly, you may need to experiment to see how well any camera works with IR illumination.