Will I be handicapped if I don't choose an alignment?

Currently, characters who do not focus extensively on either light or dark will be disadvantaged due to not being able to equip a second relic. It's not known how soon this will be fixed.

So far the only things that I have seen that is affected by your Light/Dark rating are requirements on certain pieces of armor and weapons.

Some character dialog (ie. from your companion) is also driven/derived from your light/dark choices when questing with the companion.

Aside from that, your alignment should have no major impact on your game play.

Yes, you won't be able to use items whose requirement is something like “Dark I” (or “Light I”). But I think it's only a small handicap. (Items with alignment requirement are quite rare, but you can also buy them in special dark/light shops.)

It's possible that the alignment will affect the story in some way, but I don't think I encountered anything like that yet.

There are certain items that can only be used by players above certain Dark or Light levels. The lowest level items I've seen are various Lightsabers.

In contrast to that, there are certain items that can't be used if you have over a certain light/dark level. The lowest level example of these are the various Lightsaber Color crystals. Blue and Green lightsaber crystals can't be used by anyone with Dark 1 or higher. Red and Yellow lightsaber crystals can't be used by anyone with Light 1 or higher.

So yes, there are certain benefits for remaining neutral. As a reminder, you need to have a difference between your two scores of 1000 to become Light 1 and -1000 to become Dark 1.