My Miko had a very similar issue; sometimes it was 2 days in row, then 5 days later again. He was also eating up the vomited food immediately after and going on with his day as if nothing had happened. I'm not sure why your vet is so concerned... this is pretty common for puppies. Sometimes playing too much can rile up the stomach, they vomit, eat it, and go on playing. Even if it's in the middle of the night, which is what Miko used to do (I'd wake up to the heaving sound, put him down, he'd vomit, eat it, and return to sleeping as if nothing had happened), I still wouldn't worry. My vet said it was normal, I had a second vet tell me it's normal, and I've seen puppies of all ages have this issue. It will eventually resolve itself. Miko is now 5mos (he had the issue from 8wks to 4mos old). Baloo, our second pup, is having a similar issue too (though not as constant). If she's eating, pooping, playing, and sleeping well, there's not much to worry about. Just chalk it up to puppy-ness, IMO.
I will say, however, that 5 meals a day is probably too frequent. I was giving Miko 6 meals a day when he was younger, and when I switched to 3, the problem got MUCH better. Has she been on Medi-Cal Gastro Puppy since you got her? Or is she typically on something else? The kind of food can also matter. Some dogs can be allergic to a certain ingredient in their kibble (for example, Miko is allergic to grains, so he eats grain-free kibble now). So maybe switching to a new food might help figure out if it's an allergy of some sort. If you do this (and it's not a bad idea if you want to rule out allergies!), remember to switch over slowly... 3/4 regular food, 1/4 new food, then 1/2 regular, 1/2 new, etc. over the course of 2 weeks so as to not upset the stomach even more. I'm not familiar with Medi-Cal, but here is a good way to rate the quality of your kibble:
Start with a grade of 100 points
For every listing of "by-product", subtract 10 points
For every non-specific animal source reference, such as "meat" or "poultry", meat, meal or fat, (not to be confused with actual protein source stated such as chicken, lamb, turkey etc), subtract 10 points
If the food contains BHA, BHT, or ethoxyquin, subtract 10 points for each
For every grain "mill run" or non-specific grain source subtract 5 points
If the same grain ingredient is used 2 or more times in the first five ingredients (i.e. "ground brown rice", "brewer's rice", "rice flour" are all the same grain), subtract 5 points
If the protein sources are not meat meal and there are less than 2 meats in the top 3 ingredients, subtract 3 points
If it contains any artificial colorants or preservatives, subtract 3 points for each
If it contains corn (ground corn, corn gluten, whole grain corn etc) subtract 3points
If corn is listed in the top 5 ingredients, subtract 2 more points
If the food contains any added animal fat other than fish or flaxseed oil, subtract 2 points
If lamb is the only animal protein source, subtract 2 points
If it contains soy or soybeans, subtract 2 points
If it contains wheat, or components of wheat such as gluten, subtract 2 points
If it contains "digest", subtract 5 points
If it contains salt, subtract 1 point
Extra Credit Bonus Points:
If any of the meat sources are organic, add 5 points
If the protein source is meal vs meat, add 5 points
If the food is baked not extruded, add 5 points
If the food contains probiotics or prebiotics, add 3 points
If the food contains fruit, add 3 points
If the food contains vegetables (NOT corn or other grains), add 3 points
If the animal sources are hormone-free and antibiotic-free, add 2 points
If the food contains barley, add 2 points
If the food contains flax seed oil (not just the seeds), add 2 points
If the food contains oats or oatmeal, add 1 point
If the food contains sunflower oil, add 1 point
For every different specific animal protein source (other than the first one; count "chicken" and "chicken meal" as only one protein source, but "chicken" and "fish" as 2 different sources), add 1 point
If it contains glucosamine and chondroitin, add 1 point
If the vegetables have been tested for pesticides and are pesticide-free, add 1 point
Bonus credit - If the food contains NO grains, add 10 points
Kibble's Quality Score:
100 + = A+ (Excellent Kibble Quality!)
94-100 = A
86-93 = B
78-85 = C
70-77 = D
69-below = F (FAILED - Very Poor Kibble Quality)
Grade A+ (Excellent Kibble Quality!)
Timberwolf Organics Lamb and Venison / Score 136 A+
Innova Evo / Score 129 A+
Solid Gold Bison / Score 123 A+
**** Van Patten's Natural Balance Ultra Premium / Score 122 A+
Eagle Pack Holistic / Score 119 A+
Canidae / Score 117 A+
Wellness Just for Puppy / Score 117 A+
Authority Harvest Baked / Score 116 A+
Chicken Soup Senior / Score 115 A+
Eagle Pack Holistic Chicken / Score 114 A+
Innova Dog / Score 114 A+
Natural Balance Duck and Potato / Score 114 A+
Foundations / Score 111 A+
Kirkland Signature (Diamond Super Premium) Chicken, Rice, and Vegetables / Score 110 A+
Wellness Super5 Mix Chicken / Score 110 A+
Kirkland Signature Puppy (Diamond Super Premium) Chicken, Rice and Vegetable / Score 108 A+
**** Van Patten's Duck and Potato / Score 106 A+
Royal Canin Natural Blend Adult / Score 106 A+
ProPlan Natural Turkey & Barley / Score 103 A+
Flint River Ranch / Score 102 A+
Wolfking Adult Dog (bison) by Solid Gold / Score 102 A+
Grade A
Hund-n-Flocken Adult Dog (lamb) by Solid Gold / Score 100 A
Royal Canin Bulldog / Score 100 A
Diamond Large Breed 60+ Formula / Score 99 A
Royal Canin Dachshund 28 Formula / Score 97 A
Sensible Choice Chicken and Rice / Score 97 A
Eagle Pack Large and Giant Breed Puppy / Score 94 A
Eagle Pack Natural / Score 94 A
Do you take her out a lot? At 14 weeks she's probably had her first 3 sets of DHPP vaccines (including parvo), but isn't considered fully vaccinated until her 4th set (usually at 4mos, at which time they usually give the rabies vaccine too). She might be eating little things outside that might be upsetting her stomach (some pups just have sensitive tracts). If my Miko eats just a few cat food kibbles, he'll have diarrhea shortly after.
Some other things to consider... has she been dewormed? Sometimes puppies who have worms or who are undergoing the deworming process, might vomit often. This sounds gross, but you might want to consider looking through her vomit (before she eats it up again) to see if you can find any worms. It may be as little as 1 or 2 in there. It's worth checking (I know I did!!).
Even though she's been off the antibiotics for some days now, it can take a while for the body to readjust, and vomiting every few days in this process is also typical.
Best of luck, keep us posted