I'm a little confused if he only vomited only after the soy. And has had multiple exposures to milk w/o problem.
It's possible it's getting used to the different food, a gag reflex, air bubble, a cold or allergy. The vomiting and cheeks red is a flag for me.
Regarding pp's saying that allergic reactions happen right away...that's incorrect. It can take several exposures over several days for a reaction to occur, as the person gets sensitized to the allergen. Excema, bloody stools, mucus poop would be perfect examples of that. A reaction often, however, is in relation to a recent exposure. Hives or even anaphylaxis type reactions can occur within seconds to several hours. A food protein induced enterocolitis syndrome (FPIES) reactions (my dd had) can be 2-5 hours. Fpies is an allergic reaction that results in a gastro response (profuse vomitting, often non stop, diarrhea, lethargy, shock). There's also other type of allergies like eosinophillic stuff that has different reactions. I'm not up on them though.
Feel free to look up my other allergy posts.
If you're concerned at all, you should contact a pediatric allergist. Regular peds are not up to date or fully informed of food allergies.
Don't worry if you need to do soy, as a pp cautioned. The estrogen issue has not been proven and you need to do what your child needs. My dd's been on soy for almost 3 years.
Were the soy formulas different types? Are you sure it's dairy free (casein, lactose, whey etc).
FYI for whomever - Lactose intolerance is not an allergy. Food allergies are an immune response and can be fatal. Intolerance is when one doesn't have the stomach enzymes to digest the food in lactose it's the sugar in it that's not able to be digested (b/c of a lack of lactase I believe).
TRUST YOUR GUT! If you think allergy is involved, call an allergist. We see the ped allergists at Children's Memorial (but the glenview annex). If you do try the milk, only try a teaspoon and in the morning, preferably not on a weekend. Be prepared to take action (calling 911) if necessary. Not to worry you, but to give you the tools to do what you feel is right.
Good luck!