Halachically, there is no requirement to wear a black hat while davening. The Mishna Berurah says a person should dress respectfully, as per the standard of the times. He gives an example of a hat for his times. However, nowadays, where wearing a hat indoors is sometimes considered disrespectful (such as in a courtroom, during singing of the national anthem, etc), it definitely is not required for davening.
The only reason to wear one now is to show that you identify yourself as belonging to a certain group. If you feel like yo’re part of that group, wear the hat. Otherwise, don’t. But don’t make the mistake of thinking that it’s required.