Firstly, I am anti anti-vaxxers and strongly hold they are acting contrary to halacha. Having said that, and commenting on the OP’s question with a question:
Suppose I am from the MO community. The Rabbonim in my community give various leniencies to certain halachic issues. Is there anything wrong with seeking a psak from more chareidi poskim?
Suppose I am from America, where Reb Moshe was the leading posek. Is there anything wrong with seeking out the psak of Rabbonei Eretz Yisroel that were machmir in certain areas?
Finally, suppose my community generally uses OU as the kashrus standard, would there be a problem if I seek out “chassidishe hechsherim” which have certain chumros, such as not relying on various kulos for bishul akum, checking individual tuna fish instead of spot checking etc.?
The problem of “shopping for a psak” is when your rabbonim forbid something and then you start an international search to find a Rov that will give a hetter!
I know of women that were pregnant (a healthy, normal, uncomplicated pregnancy) and their Rabbonim paskened that they must fast on Yom Kippur. They called dozens of Rabbonim in various cities, till one was maykil and permitted them to eat on Yom Kippur!
I know of girls that went to seminary in EY and their Rabbonim paskened they must keep two days of yom tov, so they shopped around till they find a psak that allowed them to only keep one day Yom Tov!