Rabbi Kaganoff says that the point is not that it is mobile but how it is used. That is, it appears that even if he drives around in it, if he lives in it permanently he should put up a mezuzah (though possibly without a bracha). It appears that the reason is that its purpose is to provide a dwelling. On the other hand, a car, whose purpose is transportation, does not require a mezuzah.
What about a Mobile Home?
The Minchas Yitzchak (2: 82) discusses whether someone who lives
permanently in a mobile home is required to put up a mezuzah,
concluding that he is required to do so; however the Minchas Yitzchak
is uncertain whether he should recite a brocha when he puts it up.
Rabbi Kaganoff mentions this concept when he discusses whether an elevator would require a mezuzah
Dayan Weiss questions whether an elevator requires a mezuzah, since it
constantly moves and cannot be considered a residence. He compares an
elevator to a moving residence, regarding which we find a debate as to
whether it requires a mezuzah. Rav Avraham Dovid of Butchatch, usually
called “the Butchatcher,” rules that a moving residence requires a
mezuzah. According to this opinion, someone who lives in a van or
truck requires a mezuzah on the door, even if he constantly drives it
to new locations (Daas Kedoshim 286:1)!
The major annotator to the Butchacher’s commentary, the Mikdash Me’at,
disagrees, contending that a moving residence is considered a
temporary dwelling and never requires a mezuzah. In a different
responsum, Dayan Weiss deliberates whether mobile homes require a
mezuzah, since people often reside in them, whereas using a bus or
automobile as a residence is considered temporary and does not require
a mezuzah (Shu’t Minchas Yitzchak 2:82; see also Chovas HaDor pg. 37).