Conversion from Simplified to Traditional Chinese

Short answer: No, not reliably+high quality. I wouldn't recommend automated tools unless the market isn't that important to you and you can risk certain publicly embarrassing flubs. You may find some localization firms are happier to start with a quality simplified Chinese translation and adapt it to traditional, but you may also find that many companies prefer to start with the English source.

Longer answer: There are some cases where only the glyphs are different, and they have different unicode code points. But there are also some idiomatic and vocabulary differences between the PRC and Taiwan/Hong Kong, and your quality will suffer if these aren't handled. Technical terms may be more problematic or less, depending on the era in which the terms became commonly used. Some of these issues may be caught by automated tools, but not all of them. Certainly, if you go the route of automatically converting things, make sure you get buyoff from QA teams based in each of your target markets.

Additionally, there are sociopolitical concerns as well. For example, you can use terms like "Republic of China" in Taiwan, but this will royally piss off the Chinese government if it appears in your simplified Chinese version (and sometimes your English version); if you have an actual subsidiary or partner in China, the staff may be arrested solely on the basis of subversive terminology. (This is not unique to China; Pakistan/India and Turkey have similar issues). You can get into similar trouble by referring to "Taiwan" as a "country."

As a native Hong Konger myself, I concur with @JasonTrue: don't do it. You risk angering and offending your potential users in Taiwan and Hong Kong.

BUT, if you still insist on doing so, have a look at how Wikipedia does it; here is one implementation (note license).

Is it possible to reliably automatically convert the text to Traditional Chinese in a high quality way?

Other answers are focused on the difficulties, but these are exaggerated. One thing is that a substantial portion of the characters are exactly the same. The second thing is the 'simplified' forms are exactly that: simplified forms of the traditional characters. That means mostly there is a 1 to 1 relationship between traditional and simplified characters.

If so, is it going to be extremely high quality or just a good starting point for a translator to tweak?

A few things will need tweaking.

Are there open source tools (ideally in PHP) to do such a conversion?

Not that I am aware of, though you might want to check out the google translate api?

Is the conversion better one way vs. the other (simplified -> traditional, or vice versa)?

A few characters lost distinction in the simplified alphabet. For instance 麵(flour) was simplified to the same character as 面(face, side). For this reason traditional->simplified would be slightly more accurate.

I'd also like to point out that traditional characters are not solely in use in Taiwan (They can be found in HK and occasionally even in the mainland)

I was able to find this and this. Need to create an account to download, though. Never used the site myself so I cannot vouch for it.