This was my life for so long. Every cold went right into the chest for a 3-4 week bout of bronchitis. The coughing was so severe that I had a horrible headache - literally holding my head in my hands with each cough to try to stop the head motion. It was worse at night or when lying down, so I tried to sleep propped up, which just made for an comfortable sleep. Exhaustion made it harder to recover of course, since most healing occurs at night. It was a vicious cycle with 3 or more cases per year, sometimes as many as five.
With a child, it's even harder because you're limited with traditional cough suppressants like codeine. Neti pots are okay if someone has sinus congestion, although not every case of bronchitis and asthma involves head congestion at all. And doing a Neti pot on a little kid is pretty much torture for them.
I don't think the Vaporub on the feet with the socks does any good. Maybe it stimulates a little warmth and then some circulation, but I don't think the Vaporub people even claim that's effective at all. Consult Snopes.com - they say it's completely unproven and stems from an article in 2002 that just went wild. Probably it was posted by someone with a store that sells Vicks!
The multivitamin he's taking probably has practically no nutritional value - most of them don't have enough for what our needs are today. There are too few vitamins and minerals, and they are based on RDIs set decades ago that are relevant for malnutrition but not for immune system support. There are ways to boost the immune system but they are not overnight and they will not help with this current case, but only build for the future.
I work in food science and I must caution you against the suggestion to add a particular individual nutrient. You'll get suggestions for more Vitamin C or more vitamin D - completely unproven, first of all. Worse, that's not how the body works - it cannot metabolize an individual vitamin or mineral without all the other "partner" vitamins, minerals, trace elements, etc., that are required for that one vitamin to do its job. So it's a waste of money (even most doctors with any nutritional background will call standard multivitamins "expensive urine" because most is eliminated, unused). And if you look on most of those products, the FDA requires a warning label to "keep out of the reach of children"! Products like Airborne that advertised immune system support were heavily fined and required to change their TV commercials for unsubstantiated claims.
There are advances in epigenetics (my field) that improve damage to the cellular machinery so that cells function efficiently. This means that good genes are turned on and bad genes are turned off - basically restoring damaged cells to their proper or original functioning before they were affected by illness, trauma, toxins, pollution, etc. The DNA (the genome) is set, of course, but the epigenome is affected by outside factors. This can lead to any number of autoimmune issues. I think that's what's going on with your son. His most damaged cells (like mine) are in the bronchial/respiratory system. There is a sustained approach that can turn that around using a food-based approach and epigenetic repair - with both an immune system support and an anti-inflammatory action. That's what got me through my history of bronchial issues and I've been bronchitis free ever since. If I stop, the bronchial tightness and coughing is back within 48 hours.
But please do not try to treat a disease with a vitamin. You can upset the balance in the body. Even if someone is "low" in a particular nutrient, you can't just throw that nutrient at him. There's a reason he hasn't been absorbing it up to this point so adding more is fruitless. And toxicity is a problem when people take individual or isolated nutrients.