There's a strong argument the LLC won't need to obtain a business license in each state you are making sales into. And these days, you don't need a physical location and may still be required to obtain a business license.
However, there are several variables at play which are state-dependent; such as what your home state is, what state you're incorporated in (which we already know), and what states you are selling into.
For example: If you were incorporated in Delaware, your home state is New York, and selling $250k of software into Illinois; the LLC may need to obtain a business license in each of these states.
A general rule of thumb would be to obtain a license in:
- State of Incorporation
- State of Domicile
- State where the LLC has any employees
- State where the LLC makes substantial sales into (e.g., over $250 or $500k per year)
The question really is one of "nexus," and most states define it differently. If you reach nexus in a state that requires you to pay income tax in such a state, then you may be required to obtain a license. Some examples of states with a rather low nexus threshold that requires licensure is Nevada and New Mexico.
Note: You'll likely be responsible for collecting/remitting sales tax as well as paying income tax on income you've earned in several states.