Reservations recommended for train from Levanto to Vernazza? (and vice versa)

The trains between those two towns are regional trains. This means no seat reservations, and therefore the trains don't sell out. Doesn't mean they might not be absolutely packed, depending on the time of day.

Not only is it not recommended, it is not even possible. You get on the train one way or another sitting or standing. It works like a commuter train. Just be sure to validate your tickets in the stamping machine before boarding.

the earlier replies pointed out that perhaps I had asked the wrong question (thank you for the assist) so let me follow-up slightly -- is it worth buying advance tickets through Trenitalia for these short regional trains between Levanto and Vernazza? As noted, we would like to leave some flexibility in case something happens either way, but would also hate to arrive at station and find that every train is sold out or impossibly crowded. Thoughts? and thanks again!

Italian Regionale trains don't have a limited number of seats. Buy your tickets at the station, from the window or from a ticket machine. In both cases don't forget to validate them by stamping them at a machine in the station hall or at the platform.
When you find your train too crowded, wait for the next one.

They don't sell out, ever! If you buy it a month in advance you have no advantage of getting on the train over someone buying a ticket out of the vending 2 minutes before the train arrives. Having said that, going Levanto to Vernazza, the train originates in Levanto so you will be the first people on the train. In Vernazza going to Levanto you might want to go on an earlier train in case it is crowded. Also, many of the IC trains to Milan also stop in Monterosso first, so you can catch that train there. And it will be a good idea to buy those in advance for the discount, as well as the seat reservation that guarantees you seats on that train.

You cannot get reserve seats on a Regionale train. Try to make this very clear --

1. When you buy a Regionale train ticket from A to B it is considered an open ticket to be used any time -- the limited used to be 60 days but that has been recently changed. So just buy both ticket to and from at either a ticket machine or a window the day of or day before. You can buy these tickets on line seven days in advance BUT your have to give a starting time then that ticket is only valid for three hours. No advantage to buying on line.

2. When ready to travel go to the station, find the platform number for the next train, find the yellow/blue/green box and validate (time stamp) your first ticket. Watch for locals doing the same thing. The clock has started and that ticket is valid for any Regionale train headed in the correct direction for the next five hours. DO NOT validate your return ticket.

3. When the train pulls in, get on quickly and in the correct class of coach. The train car will have a 2 on it somewhere identifying as second class IF the train is offering first and second class. Sometimes there is no first class coach. If no seats stand till one becomes available or walk through to the next car, and the next car.

4. When returning do the same thing.

Trying to make this simple....

Is it worth getting reserved train tickets for the short train ride
from Levanto to Vernazza since the train seems to run about every 15
minutes?

"Reserved" tickets (as far as booking a seat) are not possible for regionale trains: they do not have seat reservations. Even if you have a pre-purchased ticket, you could end up standing if the train is full. That trip takes only 8-10 minutes so standing in a crowded carriage is no big deal for that short amount of time.

Same for your return to Levanto.

So just purchase your tickets when you get to Levanto and before you leave Vernazza as a pre-bought ticket for a regionale train doesn't buy you a seat/guaranteed space. If you are concerned about your trip back to Levanto from Vernazza, just get to the station early to get your tickets and in line to board the train.

Going both ways, don't forget to validate your tickets before getting on the trains!

You can buy these tickets on line seven days in advance

This is wrong: you can buy regional trains tickets on-line up to 120 days in advance (like high speed trains), but not after the next timetable change.
The 7 days rule has been abolished 2 years ago, maybe it's time to update your files.

only valid for three hours

Sorry, but it's 4 hours all over Italy with the relevant exception of the Liguria region where it's 75 minutes (not all stereotypes are wrong after all!)

Mkalt it's true that there are no differences between buying on-line in advance and at the counter at the last minute. But the Cinque Terre stations can be so crowded at times that installIng Trenitalia's app and purchasing on-line while walking to the station is ten times easier than standing in queue. More, you're not expected to time-stamp your smartphone, one thing less to worry about!

Mille grazie to everyone who responded. very helpful information!