Is it wrong for an institution to ask for donations?

Answer:

Churches are funded solely by the voluntary contributions of their members (I Corinthians 16:1-2). No other way was laid out by God.

Man-made institutions can raise their funds in any way they desire that is legal. That isn't the problem with institutions like GBN. The problem is that they try to make their institution a quasi part of the church. For example, GBN states that it is "under the oversight of the elders of Southaven Church of Christ in Southaven, Mississippi." [https://gbntv.org/about-us/]. Elders oversee a congregation, not a business.

However, GBN doesn't just ask for contributions from Christians. They also ask churches to fund their operations. "GBN is funded entirely by donations that are given by congregations and members of churches of Christ" [https://gbntv.org/support/]. Churches cannot pay some other business to do their own work.

There is also a blurring between the role of the church and a private business. For example, the same page states, "The Lord has commanded us to reach every soul with the message of salvation." "Us" is left open and can easily be read as referring to GBN, which would be a false statement. The church is tasked with spreading the gospel (Ephesians 3:10) and so are individual Christians (Matthew 28:19-20). Where individuals may decide to create a business and use that business to spread the gospel, this is not a work of the church. There is no command by God for a church to create a business to do its duty.

See Institutionalism: A Virulent Cancer for a detailed description of why organizations like GBN are wrong.