No, do not give your bird ANY medicine until you get a proper diagnosis, as you can end-up making your bird much, much worse by giving them the wrong medication for what they are suffering from...For instance, if your bird has a Fungal/Yeast infection in his GI Tract, and you give him any type of Antibiotic, you will not only do no good, you will actually make the Fungal/Yeast infection much worse because all Antibiotics kill-off the healthy, normal Bacterial in their GI Tracts, and this is the Bacteria that normally keeps Yeast from growing in their GI Tracts...
***Most medications for any type of parasites, including "worms" in-general, are actually poisons and are very toxic. What you are describing your bird having doesn't sound like "worms" to me, it may be a parasite of some kind, but probably not worms, as the common worms that birds get in their GI Tracts will be evident in their droppings, and in their vomit. It sounds much more like he has an infection of some kind in his GI Tract, probably fungal/yeast, but you have to be sure by having a Fecal Culture done, because again, if you give him the wrong medication it can make him worse or kill him ultimately.
****Something you need to be aware of is that all birds possess an innate, natural survival instinct to hide any and all outward signs/symptoms of illness and/or pain for as long as they possibly can, sometimes for months and months. This protects them and their flocks in the wild from being targeted by predators who look for "weak" or sick birds to grab...So the fact that your bird has had "loose" or "runny/watery" droppings since you've brought him home could be an indication that he has had a GI infection since you brought him home; as I said, they can hide all other signs/symptoms from you for months and months. So now he finally is sick enough that he can no longer hide all of the signs/symptoms, and he's now vomited-up his food...(Their droppings can normally be brown or green with the white portion, which is called the "urates", and is kind of like our urine, while the brown/green portion is the feces. However, their stools being constantly or continually runny/watery is not normal...If your bird is eating a lot of fresh veggies or fresh fruit every single day, then this can naturally cause his droppings to be watery/runny, however the Vomiting is not at all normal. Usually when a bird has been having watery/running droppings for quite a while and then they eventually start vomiting, it's an indication that they are suffering from a GI Tract infections of some kind, usually Fungal/Yeast, but could also be Bacterial. It starts in their lower intestinal tract, and can actually stay in their lower intestinal tract for quite a long time, especially if it's a Fungal/Yeast infection, because they have healthy Bacteria living normally inside their intestines that keeps Yeast/Fungi from growing there. So they can grow an amount of Yeast/Fungi inside of their lower intestines for one reason or another (has he ever taken ANY Antibiotic?), and then eventually the Fungi/Yeast grows enough to overpower the healthy bacteria, and eventually it then starts to spread throughout the rest of their Gastrointestinal Tract, eventually reaching their stomachs and then their Crops, and this is what starts causing the Vomiting...
****This is an indication that whatever infection he has is progressing and getting worse and worse, and he needs to have a Fecal Culture done ASAP to determine what is going on, as well as what medication is the correct one to treat exactly what is wrong. Birds are extremely sensitive to any and all medications, and due to a Budgie being such a small, lightweight bird, you can actually kill your bird very easily by just giving him medications such as Antibiotics and especially Anti-Parasitic/Anti-Worm medications (they are poisons) without knowing if it's the correct medication to give him. Not only will it do no good if it's the incorrect medication, but the meds in and of themselves can cause severe or lethal harm to your bird. So never medicate your bird at home without having an Avian Vet Specialist run the correct tests to diagnose what is wrong and what medication will be effective.
On that note, I'm not going to challenge what the Vet did for your female Budgie with the Scaly Face Mites, but what I will tell you to do is to keep an extremely close eye on his face and especially his breathing symptoms, and if you don't see marked-improvement in a week of those treatments, OR if you see the Scaly Face Mites getting worse, or if his breathing symptoms get worse (if his wheezing gets louder or more frequent, if he starts coughing, or if the patches on his face do not decrease or they spread to other areas such as on his beak or his cere) then you need to get a second opinion with another Avian Specialist, preferably...Or, take him back to the same Vet if that's the only option within a drivable-distance of you, because chances are that the medications he's on need to be changed...
I have never heard of an Avian Vet treating Scaly-Faced Mites with any other medication but Ivermectin, and certainly not with topical medications that are applied to his face multiple times a week...A single, one-time injection of Ivermectin is all it takes to completely eliminate all Scaly-Face Mite infestations will no relapses happening. All of these drugs that are used to treat Mites are poisons, and birds being so sensitive and small should not typically be given multiple doses of poisons, nor should they be put on the bird topically because you know that they are going to be preening themselves constantly and actually then ingesting the topical med, which is a no-no...Most importantly though is the fact that a single, one-time injection of Ivermectin will totally cure the Scaly-Face Mites 99.9% of the time, and is totally safe when given in the correct dosage, based on the bird's weight...There shouldn't be any reason to make the bird endure multiple topical treatments a week and extend the length of his discomfort from the mites...So I'm perplexed as to why the Vet didn't just give her an Ivermectin injection and be done with it...
I don't know if the Vet ran any tests at all on your female to ensure that she does not have a serious Upper-Respiratory Infection, but it doesn't sound like it, and that's scary. You cannot blame a bird's wheezing and URI symptoms on a case of Scaly-Face Mites, so please, watch your bird very carefully, and if you notice the Respiratory/breathing symptoms not improving at all or getting at all worse, then please take her back and demand that the Vet take do both a Fecal Culture and Respiratory Cultures/Sinus Cultures, because if your bird actually has a UR Infection that is totally separate from the Scaly Face Mites, which would be much more common, then she's not being treated for it and it could become much, much worse.
