Hi Robin Mjelle , it seems you're being initiated into the wonderful world of anaerobic microbiology which creates a number of challenges compared to aerobic bench top work
For shaking, I would say it really depends. Some strains really don't like the physical disturbance and will actually grow slower. I found that with small enough culture volume in Balch tubes (5 mL) that shaking was not required even when growing photosynthetic bacteria that can self-shade their container. However, I have found that if you want to ensure equal distribution of nutrients, especially for cultures that can grow quite vigorously, sometimes it is necessary or all of your cells will pellet at the bottom if they're not motile.
My recommendation would be to try it both ways, what I actually found was the best for some of my cultures (which grew over 1-2 weeks scales) was just a simple inversion to gently mix once a day. This made sure cells had better access to nutrients in solution especially for larger cultures in serum bottles. It also released some of the gasses they produced lock into microbial mats that can form at the bottom of containers. Some of these gasses are quite important for growth depending on the strain (namely CO2, H2, or CH4).
What is your end goal with this strain? That will also affect the culturing protocol.
Hope this helps, always happy to help a fellow anaerobic microbiologist!