I've used in the past foam brushes to apply my water-based varnish. I'm wondering if there's other types of brush that can be used and what are the advantages and disadvantages.
Generally, besides foam there are synthetic bristle brushes and natural bristle brushes (as well as sprayers, air-brushes and rollers, which I'm not going to discuss). There is great debate over which to use for what; that debate rages from house painters to fine artists.
Most agree synthetic bristle brushes for water-base varnishes, acrylics and the like and natural bristles for oil-based mediums (the solvents can damage some synthetic bristles). For me, both types work well but have slight differences in how much paint is held, how the paint leaves the brush, etc. I find I 'get use to' how the brush handles fairly quickly.
There is the question of shape, which for most furniture folk, is less of an issue unless you are doing decorative painting, stenciling or similar work. For transparent finishes, a flat tipped brush is most common. But if you have something else that you like, great; use that. Your experience with foam will guide you regarding shapes you like (for getting into corners, for example) and widths.
While slightly off question, it is worth noting that you can tweak your finish to work with your brushes. For example, by adding more thinner (water in your case) you can create a thinner coat that dries with less visible brush strokes. This also applies to binders and in the case of paints, the pigments as well.
Synthetic brushes were originally specifically intended for application of water-borne finishes. Compared to natural hair brushes synthetics are better in one key respect: they don't soften in water. Natural bristles "lose their spring" when wet, they are still usable with waterbase finishes but they become slack and unresponsive so they don't perform at their best.
The best modern synthetics have a mixture of filaments of varied diameter and they may in addition taper towards the tip, both of which contribute to improved capillary action (helping to match the performance of natural brushes). Another feature of better modern synthetic brushes is ends that have been frayed, which mimics the natural 'flags' found on traditional hog-hair brushes.
In my experience foam brushes are not ideally suited to application of waterbase finishes, primarily because they increase the tendency of bubbles to form as you brush over the surface. Because of this I think foam brushes are best for oil-based or spirit-borne finishes which are less prone to bubble formation.
If you are doing a large area you might consider a popular method used to apply water-based polyurethane to wood floors - which is to use a "T-bar" applicator that has a thin foam or fleece-like cover.
The applicator is not intended to absorb much material and again release it the way a brush does. Instead it quickly reaches saturation and then just pushes the liquid around on a flat surface. This allows for a very smooth and quick application. Apparently the finish also comes out very even I guess due to its own viscosity and the minimal weight applied to the tool; you apply no down-pressure at all.
I doubt this could be applied to a small project but if you were doing say a tabletop it might work very well.
As a caution, it could be that a method intended for a floor would leave some imperfections you wouldn't notice on a floor, but that nonetheless could be visible on something closer to the eye. But the floor I did with this method didn't appear to have any flaws related to the finish that I could see even when I looked closely.