This may not apply to you, but I must mention it: Anyone suffering from suicidal ideation should seek professional help. These aren't the kinds of things you want to try to self-medicate away.
That said, there is a form of lithium that is available over the counter: Lithium Orotate. The amount of elemental lithium per dose is lower than you'll find in prescription lithium carbonate treatment, but you wouldn't want to attempt to self-medicate in to the prescription range anyway. Lithium has a notoriously narrow therapeutic window, and it would be too easy to dose yourself into the toxic range.
I've tried lithium orotoate myself out of curiosity for circadian rhythm modification. Although the dose is lower than that of prescription lithium, the effects and side effects were still palpable. Personally, I found no positive benefits but substantial cognitive slowing and general feelings of malaise. It's also worth noting that lithium is moderately toxic to the kidneys over time.
Having tried low-dose lithium orotate myself, I would never advocate for any amount of supplemental lithium in water supplies. Let's leave the medication to those who need it, rather than forcing it upon the entire population.
Not meaningfully. I think PragmaticPulp captured it well. You can increase the amount of lithium you consume, but mostly what lithium does is improve B12 metabolism. Regardless of suicidal ideation status, taking B12 and improving diet/sleep would be better than taking lithium. B12 is consistently and significantly lower in those with clinical bipolar than controls, so it is the real problem. Superdosing lithium ensures an ideal environment for B12 metabolism, hence the benefits.
It's available as a prescription in the US. A psychiatrist can prescribe it. There are other drugs that have less side effects that can be tried first though. I wouldn't take it without doctor supervision as it's very easy to reach toxic levels in your blood. Even while on it with a doctor you need blood work regularly. It is one thing that did finally start to help me after about 10 years of dealing with bipolar 2 (which included symptoms of not wanting to live).