The biggest needle that is used for a vaccine injection is a 22Ga which has a 0.413mm diameter hole (see attached) so, it appears that an unprotected chip could possibly be injected through a 22Ga needle, however, the most common needle is a 25 Ga. needle which has a hole of 0.26mm so the chances of a chip being injected is slim. The chip would need a coating to protect it in the hostile environment of the human body. Would 0.01mm total coating be enough? Also, there's no detail on how to find the chip and check it's datasheet or even it's availability. TBH, The smallest I've been able to find is 1.2mm and they claim to be the smallest in the world as at 2018 so I don't think that chip was able to be manufactured. It was able to be made but manufacture is a completely different ballgame.
https://www.hamiltoncompany.com/laboratory-products/needles-knowledge/needle-gauge-chart
https://immunisationhandbook.health.gov.au/resources/handbook-tables/table-recommended-needle-size-length-and-angle-for-administering-vaccines