There are numerous laws which make it illegal for governors to use taxpayer resources to promote or fund their political campaigns. This typically means that sitting governors can’t make fundraising calls from their official offices, they can't send fundraising appeals on state letterhead, they can't commit quid pro quo appointments for campaign cash, and so on and so forth.
The laws are different in every state, but the spirit remains the same.
Based on these common sense ethics laws, it would also be against the law to misuse a taxpayer funded office in order to grow a campaign email list, raise campaign cash online, or increase friends/fans/followers of political campaign social networks. Anyone who has purchased Facebook ads to increase fans or bought email lists can attest that they have legitimate and significant costs.
There are currently seven sitting governors, five Republican governors, one Republican turned Independent governor, and one Democratic governor, who are breaking the law by supporting their political campaign's online efforts with taxpayer-funded resources.