Is there a specific way to cite the unpublished procedures of a research lab, or even to use the lab itself as an information source for citation? I'm writing a mock SOP for a course that requires a reference for the procedures, but my lab supervisor only told me to "cite the lab" for an unpublished and informal list of steps for MIC determination.
It depends if these procedures common in your field.
If they are you can briefly describe them as established (or widely adopted) practices in the field or naming the well known techniques. e.g., We used sample prep technique X, measured Y with Z equipment, and used A, B, then C tools to analyse it. Another option is to cite the published method or protocol in a journal and describe how (and why) your lab modified the procedure.
This citation does not have to be reputable journal (although it is preferred), it could be a thesis from a previous lab member or a lab's website, blog, or GitHub repo as long as the source is acknowledged and the full procedure can be accessed (reproduced, critiqued, or verified). Perhaps considering putting your procedure on GitHub? We do this for both our workshop lesson resources and procedures shared within the lab.
Failing that, your other options (in order of preference) are to cite articles "in press", "unpublished work" or "personal communication". Generally this refers to early access to data that your audience should be able to access once it's been published too. These are accepted practices in many fields but generally used as a last resort when a paper can't be given. Note that a "personal communication" is generally citing a person rather than a group, in this case probably your professor. Personal communications are cited in text but not listed in the bibliography. How accepted this is depends on the person the reputation in the field of you're citing and whether you can describe how they communicated it (e.g., in lecture notes, a seminar, or course laboratory manual). For example it is clear here why the personal communication is from a reliable source:
This analysis used the method of Parker et al. (2009) designed for microarray data with the following modifications for RNASeq data (J. S. Parker, personal communication 2016).