Technically, kinda? But it probably wouldn't be very visible, and it's a bit unlikely.

So, the orbital period of an object relates to the "semi-major axis" of its orbit. The "major axis" of an elliptical orbit is the distance from when the object is at its closest point to the Sun, to the distance from when the object is at its furthest point from the Sun. The "semi-major axis" is half of this. For a circular orbit, the "semi-major axis" is just the fixed orbital radius.

Earth's orbit is pretty close to circular (although a little bit elliptical), and has a semi-major axis of 1 AU. This means that any object that has the same orbital period (one year) also needs to have a semi-major axis of 1 AU. This could be an elliptical orbit from 0.5 AU to 1.5 AU, a more circular orbit from 0.9 AU to 1.1 AU etc. At its most extreme it could go from grazing the Sun at almost 0 AU, up to almost 2 AU.

This means that the comet will have to always be relatively close to the Sun. Jupiter is 5 AU from the Sun, and Mars is 1.5 AU from the Sun, so if your maximum distance from the Sun is 2 AU, you're still very much in the inner solar system.

This is significant here because it affects how comets comet. A comet carries cold material from the outer solar system, and then warms up as it approaches the Sun. Getting heated and blasted by solar light and radiation causes outflows of dust and gas from the comet, producing a huge 'coma' tail of material streaming outwards from the Sun (note - the coma isn't a tail behind the comet, it always flows away from the Sun).

A comet that close to the Sun would be generally warm all the time, and not have such dramatic outbursts. It also would lose material every time it approaches the Sun, which means that if it was ever dramatic, it would burn out pretty quickly.

There is a comet with a period of 1.4 years - https://en.wikipedia.org/wiki/3200_Phaethon - however its dim enough that an amateur astronomer with a backyard telescope can only see it if it happens to be passing close to the Earth. That's generally what you'd expect a year period comet to be like.

Comets can have pretty regular orbits - although the outgassing of material can add a twist. But you could have a fairly inactive comet with a period of about a year, and it wouldn't radically shift. Like, if it had a period of a year and a day, it would just shift relative to the Earth's orbit by one day a year.

What you might want to look at is meteor showers. Meteor showers do regularly occur at the same time of the year. Here, there is a long tail from the disintegration of an old comet that remains in orbit around the Sun. Although the individual chunks of stuff in the tail are all in motion, they all are following the same orbit, which means that the tail as a whole stays in roughly the same place in the Solar System. This means that if the Earth passes through the tail, it will pass through the tail at the same point in its orbit every year, which is why you get meteor showers at the same dates every year. If you're looking for annual astronomical events that are visible to the naked eye, that's a common one. Some meteor showers are more dramatic than others, so there's no reason why some planet might have a particular strong on at a certain time each year.