Osteoporosis is a well-known side-effect of partial gastrectomies. Hence, the recommendation to supplement.
There is less intrinsic factor too. For a minority, this makes it harder to absorb B12. Another rare, but known side-effect that is related to nutrition is anemia.
However, most of the micronutrients are absorbed in the intestines so having a small stomach shouldn't be a big issue. I haven't been able to find any studies showing any other common nutritional issues with partial gastrectomies when I was doing my sleeve research. (There's always that one guy that everything happens to, of course.)
I searched on partial gastrectomies because there is more data on them. The problem with that is that these partial gastrectomies are often done on people with a normal weight and/or the pylorus valve gets removed. That makes them slightly less applicable to us. But it's the best data I've found. (I found 6 pages of results -- about 25% of them relevant when using "partial gastrectomy" and 4 entries, none of them relevant when using "vertical gastrectomy" as my search term.)
Here is an example of the sort of study I found:
[Nutritional evaluation in patients with total and...[Rev Gastroenterol Peru. 2008 Jul-Sep] - PubMed Result
CONCLUSIONS: Malnutrition was not an inevitable consequence of a gastrectomy and may be prevented. The incidence of anemia in this group of patients has high, while other biochemical parameters such as albumin level and total proteins were not so affected.
This study included people with Total Gastrectomies and they were the ones with the most problems, btw.