You sorta can. The difference in your scenario is that your neighbor doesn't need you to mow their lawn, but Reddit requires moderators in order for the business of Reddit to function.

Here is a guide published by the state of California about whether someone should qualify as an employee of a company. Read through the first couple pages of checklists and ask yourself if a moderator fits the criteria they're looking for.

For the first 3 questions, a "Yes" answer is an indictator that the person is an employee.

1. Do you instruct or supervise the person while he or she is working?

• I would say that likely counts as a yes, because moderators have a code of conduct which is mandated by Reddit, and they must follow it in order to keep their jobs.

2. Can the worker quit or be discharged (fired) at any time?

• Reddit does not have protections in place for moderators, who can be removed from their positions at any time. Likewise, moderators can walk from their job at any time.

3. Is the work being performed part of your regular business?

• This is definitely a yes, because Reddit relies on subreddits for its business, and subreddits require moderators.

For the next 3 questions, a "No" answer indicates that the person is an employee and not an independent contractor.

4. Does the worker have a separately established business?

• This is a bit of a gray area. For the majority of moderators, this would be a no at surface value, but some subreddits that concern a specific product/company sometimes have representatives from that company on the mod team. However another criteria of this category is that moderators have the ability to add/remove other moderators at their discretion, which is an indicator that they qualify as independent contractors and not employees. Should this go to trial, this will be an item that is argued.

5. Is the worker free to make business decisions which affect his or her ability to profit from the work?

• This would likely be a no for most moderators. To expand further, one of the example criteria is whether the individual is free to utilize their own tools/resources to do their work, and Reddit limiting API access is specifically one example of this not being the case. But if the subreddit is a front for a business (as in, the subreddit's primary purpose is to sell/support a paid product or service), it likely would not qualify.

6. Does the individual have a substantial investment in their job which would subject him or her to a financial risk of loss?

• Similar to the above, I think this would be a no for most moderators. Reddit controls the platform and dictates what resources moderators are/aren't allowed to utilize when doing their jobs, so there is no independent financial investment from the moderators that is at risk.

It's not cut-and-dry, and I think that's what might make this difficult to take to court, but the argument certainly exists and the case could at least result in better terms for how Reddit must work with their moderators.

It's a bit more complex than that. Reddit hires staff to do moderation. If moderation was done solely by users, and never by paid staff, your analogy would hold more water. However, because there is a mix of paid and unpaid labor doing the same tasks, there is enough gray area that a court could weigh in either direction (although I think it is unlikely that one would find for the mods, personally).
A better analogy would be that reddit had a landscaping business, and hired some workers to do landscaping, and you just tagged along and did unpaid work for several years. Sure, the owner did tell you he wasn't ever going to pay you for your work, and you agreed to that. But the owner sold and profited off the labor you provided alongside his paid laborers. He did this knowingly.
There may be a case there.