We are planning to fix the flooring of the Laundry Room. Whenever the washer runs, you can hear and feel the vibration from the 2nd floor.
The washer is a front loader which has a high rpm.

I already balance the adjustment underneath the washer to make it level and balance the load of the laundry.

The floor is currently linoneum and we are planning to make it porcelain tiles. The room is just 6 feet by 8 feet.

Is there a way to lessen the vibration noise from the laundry room?

If anything, the ceramic tile will make the problem worse.

What you are getting is vibration transmitted through the house structure. What you want to try to do is isolate the washing machine from the structure. Usually, things like that are done by placing the machine either on springs, or on a resilient pad of neoprene or similar rubber compound. The springs are impractical, but the pad may be easily done.

I would consult with the manufacturer of the washing machine to see if they have, or know of, a product that will isolate the machine. If that fails, try a company that sells and installs the same machine.

I am not familiar with neoprene flooring.
Is it same as congoleum?
Or Is it the foam underlayment used in laminate flooring?

Can I reduce vibration similarly by gluing a foam underlayment on the plywood subfloor then just glue above it the tiles.?

It isn't flooring, it's an isolation pad, which goes between the machine and the floor. You would install your tiles first, and then place the pad over them and set the machine on it. You want to completely isolate the machine from the structure, and your tiles will become part of the structure.

I used Google (isolation pad for washing machine) and found the following link, and there were others.

http://www.kellettent.com/vib_isol.html

I would try a set of pads for your machine I have tried several having a very similar problem. I tried the Kellett ones and another set. Probably the best set of pads I have experience are from a company called “The Vibration Solution” they make Silent Feet (www.thevibrationsolution.com), great for isolating that high frequency vibration. The reason I liked their pads more is that they use a material that also isolates things for the military NASCAR and NASA (browse around on their page for a sec). But their material is better because unlike a lot of the competitors that use a “Styrofoam” like material that will break down during uses and loose isolation properties the Silent Feet will not. Worth giving it a shot hope that helps