Your question:
“However, given that my other unexposed volunteers are equally healthy
- wouldn't this be a fair comparison? Is there still volunteer bias here?”
The comparison of exposure versus non-exposure among the volunteers yields an unbiased estimate of the effect of exposure among people who are like the volunteers (internal validity) but the findings may not be generalizable to people who not like the volunteers--lacks external validity. For example, if volunteers are more healthy than the general population (they usually are) and the cohort study comparing exposed with non-exposed among volunteers finds no effect of exposure, one cannot conclude that there is no effect of exposure in the general (less healthy) population—there is (potential) bias.
EXPLANATION
An epidemiologic cohort study is a non-experimental (observational) study design which assesses the association between one or more exposures and the development of (or mortality due to) one or more diseases. The goal is to draw causal inferences about the effect of exposure and the development of (or mortality from) the disease.
As explained in this excellent 2010 review article by Song and Chung (and elsewhere):
“The hallmark of a cohort study is defining the selected group of
subjects by exposure status at the start of the investigation. A
critical characteristic of subject selection is to have both the
exposed and unexposed groups be selected from the same source
population.” [bolded for emphasis]
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2998589/
Song JW, Chung KC. Observational studies: cohort and case-control studies. Plast Reconstr Surg. 2010;126:2234-2242. doi:10.1097/PRS.0b013e3181f44abc
People in cohort studies are seldom completely representative samples of the population for many reasons. It is difficult to identify representative samples. Some people identified as part of a representative sample don’t want to be in the cohort study. Thus, in practice, people in a cohort study where the study interacts with the people in the study are always volunteers.
NOTE: Some cohort studies involve only computer record-linkage and this can be done (with approval by an Institutional Review Board) without interaction of the study with the people whose records are used in the study. Thus, there is no issue related to volunteerism.
Famous cohort studies such as the British Doctor’s study, the Framingham Study, and three iterations of Nurses’ Health Studies all enrolled only volunteers.
These links are to articles about the history of these landmark cohort studies and how the people in the study entered the study as volunteers.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5298160/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4159698/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4981810/
There are numerous other examples.
Use of a non-representative sample, for example, volunteers, in a cohort study affects the external generalizability (also called the external validity) of the results. For example, nurses who volunteered to be in the Nurses’ Health (cohort) study cohort were preponderantly white and their educational attainment was higher than that of all women in the United States. The findings of the study about the association of an exposure with a disease association might not be generalizable (applicable) to non-white women or women with educational attainment different from nurses.
See Bao et al. (cited above) for discussion of generalizability in the Nurses' Health Study I.
An important source of bias in cohort studies arises because of loss-to-follow-up where there is differential loss-to-follow-up comparing the exposed and unexposed.
See this classic (pay-walled) paper.
Greenland S. Response and follow-up bias in cohort studies. Am J Epidemiol. 1977 Sep;106:184-7. doi: 10.1093/oxfordjournals.aje.a112451
A benefit of enrolling volunteers in a cohort study is that the volunteers are (presumably) interested in the study and thus are less likely to be lost-to-follow-up.