Hello from JustAnswer.
First, it is not necessary to do anything while you are traveling. There is no urgent need to treat warts, and if you would rather not deal with the difficulties of treating warts, then it is fine to wait. In particular, if you are walking a lot as part of your traveling, you may find it preferable to wait until the traveling is done to treat the warts on the bottom of your feet.
However, if you want to treat the warts without seeing a doctor, there are several options available over the counter. The over the counter treatment that I recommend as first line treatment would be the freezing products. There are several brand names on the market. These products are not as effective as freezing done by your doctor, but it is reasonable to try at home (or wherever you are staying while traveling). It also may help to use a product that contains salicylic acid, which is also available as several brand names.
The issue with treating warts on the feet is that any intervention that damages wart tissue will tend to cause some localized swelling. And while this swelling is not an issue for most locations on the body, a small amount of swelling on the bottom of the feet feels like you are walking with a pebble in your shoe. It already may feel like a small pebble from the wart, but it will usually feel worse before it feels better if the wart is treated.
If you do not want to treat using over the counter products, then it would be fine to seek a Walk-In Clinic, but many of these clinics are not designed to treat warts, so you should contact the clinic before arriving to be certain that they are able to treat the warts.
Is there any additional information that would be helpful?