What state do I need to file taxes in for my online business?

Delaware Division of Corporations > Annual Report and Tax Instructions says you owe feed and a franchise tax. The page links to a calculation method:

All domestic corporations formed in Delaware are required to file an Annual Report and to pay a franchise tax.

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Although Limited Partnerships, Limited Liability Companies and General Partnerships formed in the State of Delaware do not file an Annual Report, they are required to pay an annual tax of $250.00.

Corporations are taxed in the state(s) of nexus, so you'll have to file taxes in each state with a "connection." This page, US Business Law / Taxes > Nexus (Tax Nexus), has an excellent overview of what the corporation is responsible for paying.

Nexus is determined differently for income taxes and for sales tax purposes.

In general, nexus is created for income tax purposes if an entity derives income from sources within the state, owns or leases property in the state, employs personnel in the state in activities that exceed "mere solicitation," or has capital or property in the state.

Nexus is determined for sales tax purposes more loosely. [...] physical location [...] resident employees [...] property (including intangible property) [...] [presence of] employees who regularly solicit business

For specific obligations in each state, take a look at SBA.gov's "Learn About Your State and Local Tax Obligations" page. It's an excellent resource!