I'm assuming that your business is a sole proprietorship, and that you have not set up a corporation or an LLC.
The DBA form (Assumed Name Certificate) that you submit to a Texas county just registers the name that you are using for your business. It does not really create any type of legal entity. If you have not created a corporation or LLC for your business, then you have a sole proprietorship, which means that you report your business's financial information on Schedule C of your personal tax return.
This is important to understand for your question, because the ITIN, or Individual Taxpayer Identification Number, that you need to fill out your tax return does not belong to your business, but to you personally. It is really not directly related to your business.
If you don't already have an ITIN for yourself, you need to use Form W-7 to apply for the ITIN. This is typically done by sending in the Form W-7 with your first tax return. From the Form W-7 Instructions:
When To Apply
First-time ITIN applications.
If you’re applying for a new ITIN, complete and attach Form W-7 to your tax return when you file the tax return for which the ITIN is needed.
You won't be able to do this online. According to the instructions:
You can’t electronically file (e-file) a return using an ITIN in the calendar year the ITIN is assigned. If you need to file multiple-year returns, you can attach them all to your Form W-7 and submit them to the IRS. Once your ITIN is assigned, you can e-file returns in the following years. For example, if you apply for and receive an ITIN in 2020, you may not e-file any tax return using that ITIN (including prior-year returns) until 2021.