Travertine is a marble, and as such, is more likely (like all marbles are) to stain, change & develop a patina over time--especially if exposed to soaps, salts, chemicals (in cleaners, make-up, skin care products) and water. Polished Marble surfaces typically will dull over time and develop obvious water spots & stains from use WHEN subject to water in high-use areas like baths & kitchens.

Go to the marble institute to read more about marbles & travertines.
http://www.marble-institute.com/

There are fillers for marbles, but they are NOT as durable as the stones and granites themselves.

Instead of trying to do it yourself, you might be better off:
A) contacting a professional stone fabricator/installer to fill the travertine on-site,
B) changing the vanity top to a different material or
C) living with the idea that filled travertine/marble may lose some of the filling material over time --especially in a high use area like a bathroom .
( If you bought what was suppose to be filled travertine, but it arrived not fully filled, you should have rejected it upon delivery-- unless you can get the company who sold it to you to fill it more. Generally, unfilled marble is less money than filled marble. The filled areas of the marble with use & over time will never behave the same as the natural stone itself. Some imperfections are part of what one gets with ALL natural materials-- like wood, bamboo, stones, etc.)

The pro stone installer/fabricator should be able to fill, buff and seal your travertine. But they may NOT be able to restore its finish to the way it looked when it was first installed if it has had a lot of use or was installed many months ago. (That is an inherent downside of marble.) Some crevices/voids are a part of marble just like grain variations and wood knots are a part of how many woods look.

You can slow down stains by sealing the marble every 6-10 weeks, and taking care not to spill makeup, soaps, chemicals, water on the surface. But how practical is that? (It's also why marble floors in high traffic areas are a BAD idea in climates like the upper Midwest, East Coast and areas that experience the four seasons & radical climate changes -- 80 degrees to snow & 0 degrees.)

A good architect or designer SHOULD have warned you (or the original purchaser) about the pitfalls of marble in bathrooms and kitchens (and areas prone to standing or constant water. ) I always warn my customers before they buy marble. The Big Box home stores & contractors selling marble tops for bathrooms often do not know nor advise customers about the pros & cons of systems and materials (because they often do not have expertise in those areas.)

A professional marble installer/fabricator should be able to fill the crevices in your travertine. I would imagine a curing time and sealer would be required (so you might have to fore go use of the vanity for more than 48 +/- hours or as required.)

I do NOT recommend you install glass over the vanity, as that has all kinds of practical problems (like cutting around the faucets, polishing the opening for the sink). Anytime you have glass in a bathroom, it needs to be tempered glass (for safety reasons, or you can kill or badly injure yourself if it shatters for some reason. Often this is also a building code requirement that the glass be tempered or safety glass.) Plus, it kind of defeats the purpose of having travertine if you cover it (even with glass. It would look like a low-rent, amateurish "solution" if you "McGyvered" it with glass.)

Also, the filler applied to the marble CAN STILL STAIN or discolor. The sealers only slow stains & water spots down. Sealers don't make the marble stain-proof. If you don't like the patina changes or the pores in marbles, then travertine (and some granites) may not be for you.

Depending on the size and location of your sink, you might be able to find a pre-fab granite or Corian vanity top for a lot less than a custom cut vanity top.

If you wanted a more impervious material (to water stains and the like) then the better material for you might be glass, granite or a less porous synthetic like a Corian (a petroleum derivative). But these materials have different aesthetic qualities and costs versus your travertine.

Do the research before you commit to a direction.

Pits and fissures are natural in travertine tops and floors. If the holes bother you be careful what you use to fill them. Travertine is also naturally an absorbent material and if you use colored fill the colors may bleed where you don't want it to be and ruin the look of the stone.

Anytime you are working with natural stone materials you should be careful to products designed for stone. Many chemicals in regular cleaners can damage stone tops in the long term.

There are companies out there that make products for this purpose. Check out www.mystonecare.com. They have products designed for this exact application. You may also look in the tile dept in your local Home Depot. They may also have something for you.

Here are a list of resources that will help you in your bid to fix the holes in your Travertine Vanity top. This site recommends sanding it and then using a product called Gel gloss, with it you may be able to restore some of the sheen, by applying the wax product known as Gel Gloss. I'm no expert but do check out their site you might find something helpful,Also check out the forums listed. I hope that you find something helpful.

You can use wood putty. Buy the kind that is tinted to match the finish of the wood. Let it dry then just use typical furniture polish over the top of it.

Also, you could purchase a thick piece of glass custom cut to fit over the top. This will protect the surface, its removable, and it requires no permanent change to the actual piece of furniture.