Not to be too blame-y, but I haven't seen JD around in some time. According to the page of record, he has made one action on this site in the last six months.
Can we fire a moderator?
When is it appropriate to do so?
Can we fire a moderator?
Anyone can raise a complaint against a moderator.
Community Managers can remove moderators.
If enough other moderators on the site agree a moderator can be removed.
This process is more for a contested case rather than lack of activity though.
When is it appropriate to do so?
When the moderator is no longer a net benefit to the site.
It would always be done after attempts to contact the moderator in question had either failed or received no acceptable response.
That one I'm not going to comment on.
Could be a stint of Real Life: some change in circumstance, loss of job, change of job, new baby, died. We (out here in Nonmoderator Land) don't know.
Mods at the very least ought to have basic contact info for each other and in the case of SE, an appropriate member of Staff. Goes without saying, but sometimes these things need to be said.
Before considering the firing of a Mod, I should hope that someone contacted that person to find out if something's going on.
Hypothetically, I wouldn't consider firing a Mod except under very rare circumstances: abuse of a forum member; abuse of authority; abuse of SE staff. Being absent doesn't quite reach that level. Unless there is some kind of written policy on Moderator tenure & requirements of office.
I'd like to specifically address the issue of "Should we?"
At the risk of sounding mean-spirited (and possibly even appearing to bear sour-grapes, since I participated in the last mod election and lost), we, the participants on this site, elect moderators. It isn't a birthright. There isn't tenure. That privilege comes with obligations that we, the non-mods, depend on (whether we know it's happening or not) and have the right to expect. The most common among them is frequenting the site to use those privileges (and may I point out that L.Dutch is a wonderful example of just that. When he won the last election, he took the bull by the horns and never looked back. Cheers, mate!).
I don't know JD and have nothing personally against him. But he was quite literally elected by the community based on the belief that he'd volunteer the necessary time to do the job.
I agree entirely that it would be considerate to reach out to him (and any mod who's dropped off the radar) to see what's going on. But in the case of a 6-month abstention, that process should take 1-2 weeks max and then a new election occurs. To be honest, I'm a bit surprised that SE doesn't have an automatic system to remove mod status from people who haven't used mod privileges meaningfully in some identified period of time.
Therefore, yes, we should take action to either bring JD back or remove the mod status and pass it to another.
And to keep this response as unbiased as possible: I will not throw my hat in the next moderator election.
I don't find it necessary to dismiss moderators for inactivity. That said, there is a more serious issue involved.
Do we need more moderators? If you feel that more moderators would make the site work better, you should explain why that is so (preferably in a separate question). We could always hold a moderator election and add one or more moderators. Or the existing moderators might find another way to fix the problems that you are noticing.
As far as I know, there is no limit on the number of moderators that we can have. Stack Overflow has many more than Worldbuilding does. If more moderators would make the site better, we can have them. This is true whether we kick out the existing moderator for inactivity or not. So I would recommend concentrating more on whether we need more moderators and less on potential inactivity of existing moderators.
In terms of whether or not we should dismiss moderators, I'm of the opinion that the real question is whether or not moderation is a perishable skill.
As I understand it (and correct me if I'm wrong), 2x moderators are elected at each election, and moderation is bestowed upon a person more or less as a lifetime appointment, or at least there is no clearly identified term that I could see when I voted in the last election. In practice, that means that if all moderators were always active, the number of moderators would increase (ideally lightening the load or keeping the current load as traffic on the site increases).
If this is all the case, then the concern about making time available for moderation tasks is valid, but one can't reasonably expect that time to be applied constantly forever. I'd argue that the moderation time investment should only need to be guaranteed from election to the next election intake. After that, if you aren't as active for whatever reason (including real life concerns, like family changes, career issues, illness etc.) then it's not as big a concern.
This is more a general answer, and I don't intend this statement to reflect JD's specific circumstances, or even to argue that first termers are subject to special rules by comparison to longer serving moderators. I merely point out that if there is a time commitment on a permanent investiture, the time commitment may be impractical to be permanent as well.
If moderation skill is perishable (and I'd argue that it's not) then sure; there's an argument to be made that lack of practice hours makes someone less effective in the role and that has to be considered. That said, could JD come back tomorrow and be just as effective in his role as he was before he left? Probably. In point of fact, he may come back even more effective because of some more life experience.
I would therefore argue that dismissing any moderator for absenteeism is perhaps at best unkind, at worst cutting off our noses to spite our faces given that he can be just as effective at a future time when he's ready to commit more time to us.
In full disclosure, I don't intend to stand for moderator election because I don't want or need that kind of responsibility and I'm very happy with my votes only taking effect as part of a chorus of like minded voices. But, I still respect those who do take on the role and I understand that this can be an issue for many who work doing it every day and whose workload is now greater as a result of someone missing. But like I say, if that's becoming a larger issue, the solution is to intake more moderators at each new election, not dismiss the ones we already have.