Hi Laura, and welcome.
I recommend you check out the link below to see your visa options. Unless you have family ties to Ireland, or have some in-demand, highly skilled qualifications, it can be difficult. There are visa options for education, but I'm guessing that's not what you want.
http://www.citizensinformation.ie/en/moving_country/moving_to_ireland/working_in_ireland/coming_to_work_in_ireland.html
Even though your husband works remotely, he'd still need to be eligible to work in Ireland, to live there while doing that.
Are there other specific questions you need help with?
Liam

I guess my question is: would he be eligible to work there? We don't fall under any category they have listed for "coming to work in Ireland" so I don't know if our unique situation would qualify for a different type of visa.
Thanks for the link,
Laura

Hi Laura,

Does your husband's employer happen to have an office in Ireland? If not, i think you're only option would be to prove that you can financially support yourselves during your time in Ireland. Or as Liam mentioned, go back to school and be on an education visa.

Without knowing much about your husbands background, Laura, it's almost impossible to comment. If he is a non-EEA citizen, then he will need a visa of some sort. That may be employer-sponsored, or some other kind, but it's not easy to just come and work for non-EEA citizens.

no, unfortunately there is no office in Ireland. Who would we contact in order to prove that we could financially support ourselves?

Liam, we are both US citizens.

I think what Kate is referring to is the Stamp 0 process, which does not entitle you to work while in Ireland, even remotely. It also restricts you from use of irish public services, which most likely would include school for your kids too. It's not intended for long term immigrants. For that you have to qualify for a visa.

Is there an embassy near where you live that you could call or drop in to? There's a good chance they'll refer you to the online material for visa applicants, but sometimes it's good to talk to someone.

I know this does not sound very encouraging, but it is difficult to move to the EU with citizenship and/or sponsored employment.

You're right, a Stamp 0 wouldn't work for us. We plan on paying taxes and utilizing services (ie. school).
Yes, we did call the embassy and they referred us to a 1-900 number which cost $2-something a minute, so we decided not to do that.
It does seem like this may not work for us, unfortunately:(

My recommendation is look at your family history. It is easiest is grandparent or more recent immigrated to your current country for either of you. Some countries it can be based on years and in some cases even several hundred years like Lithuania.

I have already looked at my family ancestry. I am a quarter Irish through my paternal grandmother, but she was born in the US. If only they went back a little further...