What voltage are the lights designed for? We need to know a lot more than just the wattage rating.
Yes, more info on lamps is needed. If they use simple resistive current limiting and (assuming they are white LED) if you use four in series, 12v supply may not be quite enough to light to full brightness, three might work.
In general a voltage-regulated dc wallplug does not provide current regulation required by LEDs; not to say it can't be used under
the right conditions. Adapter itself seems pretty ordinary, except the terminal block. I find 12v/1A regularly in 2nd-hand stores for $2 each over the counter.
Even this is overkill for 1W worth of lamps. There exist very small 12v/0.25A adapters generally to charge batteries in equipment.
I've been using small flange-based LED bulb replacements for years, they can work well. One variant is 1-2 cell which uses step-up
circuit in the base. I would be careful putting these is series. Other 3-4 cell variant seems to only use a dropping resistor but higher
input voltage requirements.
Dave