I will be traveling to Hawaii soon. I'm 18 years old and only have my school ID

The purpose of the ID card is to demonstrate that you are the same person as the holder of the ticket, and thus to be allowed access to the departure area.

As Marcel notes, there are a variety of different types of identification that are accepted. For example, even if you do not drive, you can apply for a Texas ID card from a Driver License Office. Since you may need to present such a government-issued photo ID for a variety of other things in life, from checking into some hotels or even applying for a job, it'll be useful to obtain one.

Currently, if you cannot present any kind of approved identification, or if you forget your ID at home, you can still travel by following the instructions at the bottom of the TSA identification page:

In the event you arrive at the airport without valid identification, because it is lost or at home, you may still be allowed to fly. The TSA officer may ask you to complete a form to include your name and current address, and may ask additional questions to confirm your identity. If your identity is confirmed, you will be allowed to enter the screening checkpoint. You may be subject to additional screening.

You will not be allowed to fly if your identity cannot be confirmed, you chose to not provide proper identification or you decline to cooperate with the identity verification process.

Similarly, the TSA blog says

If you’re 17 or younger, no problem… you don’t need ID to travel. If you’re 18 or older, no worries… you can still travel.

How so? Simply approach the travel document checker and let them know that you don’t have your ID. …

If you’re willing to provide some additional information, we have other means of substantiating your identity, such as using publicly available databases. If we can confirm your identity, you’ll be cleared to go through security, and you may or may not have to go through some additional screening.

Your school ID may be helpful in this case.

Here there are the documents that you could use to travel in US:

Adult passengers 18 and over must show valid identification at the airport checkpoint in order to travel.

• Driver's licenses or other state photo identity cards issued by Department of Motor Vehicles (or equivalent)
• U.S. passport
• U.S. passport card
• DHS trusted traveler cards (Global Entry, NEXUS, SENTRI, FAST)
• U.S. military ID (active duty or retired military and their dependents, and DoD civilians)
• Permanent resident card
• Border crossing card
• DHS-designated enhanced driver's license
• Airline or airport-issued ID (if issued under a TSA-approved security plan)
• Federally recognized, tribal-issued photo ID
• HSPD-12 PIV card
• Foreign government-issued passport
• Canadian provincial driver's license or Indian and Northern Affairs Canada card
• Transportation worker identification credential
• U.S. Citizenship and Immigration Services Employment Authorization Card (I-766)
• U.S. Merchant Mariner Credential

Source: TSA

I will be going to Honolulu, Hawaii from Texas soon. I don't have an identification except for my school ID, but I'm 18 years old. Will I be allowed through security and be able to board a flight?

Maybe.

From https://www.tsa.gov/travel/security-screening/identification (emphasis mine)

In the event you arrive at the airport without valid identification, because it is lost or at home, you may still be allowed to fly. The TSA officer may ask you to complete a form to include your name and current address, and may ask additional questions to confirm your identity. If your identity is confirmed, you will be allowed to enter the screening checkpoint. You may be subject to additional screening.

You will not be allowed to fly if your identity cannot be confirmed, you chose to not provide proper identification or you decline to cooperate with the identity verification process.

Do you want to bet your trip on a maybe? do you want to risk getting stuck in Hawaii? if not then you should go and get yourself an approved form of ID before your trip.