Do people "subvocalize" during thinking or "internal monologue"?
Short answer
Sub-vocalization can also be detected during cognitive demanding tasks other than reading.

Background
As far as I can see, most literature acknowledges sub-vocalizing as being only detectable by measuring the electrical nerve activity that drive the speech organs (e.g., Parnin (2011) and NASA Research). When people perform complex tasks, these sub-vocal utterances can be recorded as electrical signals sent to the tongue, lips, or vocal cords, and these can be detected. 

Sub-vocalizing has been mostly described in terms of reading (e.g. Slowiaczek &amp; Clifton, 1980). Nonetheless, more recent literature also associates and acknowledges sub-vocalization when people perform mental tasks that claim a lot of mental processing, such as software programming (e.g., Parnin (2011), so the answer to your question is yes, it also occurs during thinking.

Interestingly, as a side note, stutterers sub vocalize more slowly, reflecting the fact that indeed much of the same mechanisms are at work during sub-vocalization as during normal speech (Bosshardt, 1990). 

References
- Bosshardt, J Speech Hear Res (1990); 33(4): 776-85
- Parnin, IEEE 19th International Conference on Program Comprehension (2011)
- J Verbal Learning Verbal Behav (1980); 19(5): 573-82