The data is all over the place. There's this peer reviewed study stated:
> "By analyzing viral loads of over 16,000 infections during the current, Delta-variant-dominated pandemic wave in Israel, we found that BTIs in recently fully vaccinated individuals have lower viral loads than infections in unvaccinated individuals. However, this effect starts to decline 2 months after vacci- nation and ultimately vanishes 6 months or longer after vac- cination. Notably, we found that the effect of BNT162b2 on reducing BTI viral loads is restored after a booster dose. These results suggest that BNT162b2 might decrease the infectious- ness of BTIs even with the Delta variant, and that, although this protective effect declines with time, it can be restored, at least temporarily, with a third, booster, vaccine dose."
https://www.nature.com/articles/s41591-021-01575-4.pdf
However, we also know that the viral loads are also very high in elders, in the obese, comorbidity as well as males. The drop in effectiveness happens the fastest in males, elderly and those with comorbidities.
Also expecting everyone to keep getting boosters every 2-6 months (Canada is allowing after 2 months) is entirely unreasonable.
There's other studies which found same viral loads in vaccinated vs unvaccinated even in mild cases (though it's not clear after what period did they perform the study).
Highly vaccinated Dane County in Wisconsin, US found no difference in viral loads when comparing unvaccinated to vaccinated infections, the viral load was consistent with the ability to shed infectious viruses.
https://www.medrxiv.org/content/10.1101/2021.07.31.21261387v...
Similarly, data from 36 Wisconsin counties found that both vaccinated and unvaccinated can transmit the infection. Infectious virus is found even in vaccinated when specimen Ct values are low, meaning mild cases in vaccinated individuals can spread the virus too.
https://www.medrxiv.org/content/10.1101/2021.07.31.21261387v...
Houston Methodist Hospital of Texas also found similar viral loads and low Ct values in vaccinated vs unvaccinated infections.
https://www.medrxiv.org/content/10.1101/2021.07.19.21260808v...
Similar finding in Massachusetts:
https://www.cdc.gov/mmwr/volumes/70/wr/mm7031e2.htm
Overall, there's no logical reasoning to not allow tests as a reasonable accommodation and not testing both vaccinated and unvaccinated if the goal is to prevent the spread of COVID.
Also these minute differences in viral loads for maybe 1 to 2 months after inoculation won't be sufficient to justify discrimination based on perceived disability.