Is it permissible for a person to appoint someone to perform Hajj on their behalf due to their inability to ride in a car because of severe dizziness and vomiting?

It is not permissible to deputize someone to perform the obligatory Hajj on behalf of a living person who is able, but it is valid in two cases: on behalf of the deceased, and on behalf of a living person who is permanently incapacitated, such as someone who is chronically ill or very elderly.

As for voluntary Hajj: there are two opinions, and the correct one is that it is permissible to deputize someone to perform it on behalf of both the able and the unable. However, it is not permissible to deputize someone for a voluntary Hajj for a person who has not yet performed their obligatory Hajj, despite being able to do so.

What the questioner mentioned about experiencing dizziness and nausea when riding in a car is not a valid excuse to authorize someone else to perform the obligatory Hajj, because it is something that can potentially be overcome and treated.