To verify you're the owner of your account Yahoo will allow you to verify your identity using your phone number or alternate email address.
If these were not already configured you may be out of luck.
Try using another device. If you're on a phone, try a computer. If you're on a computer, try from your phone.
When you get home, please take a few minutes to turn on 2 step logins, and update the alternate contact info in your account settings.
Without this information, in addition to problems while traveling, you're at risk of losing your account and not getting it back if it gets hacked.