Do travelers have to do qurbani?

What the schools say

• Hanafi: qurbani is wajib (required) for an adult Muslim who has the nisab amount of wealth and is a resident during the days of sacrifice. A musafir (someone on a journey that qualifies for shortening prayers) doesn't have to do it, though it's still rewarded if he does.
• Maliki, Shafi'i and Hanbali: qurbani is a sunnah mu'akkadah (strongly encouraged sunnah) for anyone able to afford it, and they don't drop it for travelers. The Maliki school does exempt pilgrims performing Hajj.

So under the majority view, travel doesn't change much: it's encouraged, not obligatory. Under the Hanafi view, it's obligatory only at home. See What is qurbani and do I have to do it?

The Prophet ﷺ sacrificed while traveling

Thawban, a freed servant of the Prophet ﷺ, said the Prophet ﷺ offered his sacrifice during a journey and then told him to prepare its meat, and Thawban kept feeding him from it until they reached Medina (Sahih Muslim 1975). This shows that being away from home is no barrier to qurbani.

How to do it while you're away

• Book online. Most Muslim charities in the UK, Europe and the USA let you pay for a qurbani that's done abroad and the meat given to poor families. See Can I pay for qurbani abroad instead of locally?
• Ask family at home. Someone can do it on your behalf if you give them permission and the money.
• Check the timing. The sacrifice is done after the Eid prayer, from the 10th of Dhul Hijjah up to the end of the days of sacrifice (scholars differ between the 12th and the 13th).

Pilgrims are a separate case

If you're traveling for Hajj, the animal pilgrims offer for tamattu' or qiran Hajj is called hady, which is a Hajj rite, not the Eid qurbani. Whether a pilgrim also does qurbani for Eid is a separate question; ask your Hajj group's scholar. See What is the difference between Hajj and Umrah?