Yes he did. In the Book of Acts, we do not see a categorical reading that Paul asked his supporters for money to sustain his missionary work. We see in chapter eleven that the disciples raised money to help their brothers in Jerusalem during a time of famine (Acts 11:30). In another part, we see Paul receiving support from the Macedonian churches during his missionary work in Corinth (Acts 18:5 and 2 Corinthians 11:9). In chapter 13, we understand that Paul and Barnabbas were sent off to missionary work by the Church of Antioch (Acts 13:3). We can only assume that being sent off also means being financially supported by brothers and sisters from Antioch.
But did Paul asked for money? Did he go around telling people that his funds are low and he needs more financial support in order for him to continue in his missionary work?
In other parts of the Bible, we see that Paul asked for money to help other people. He implored the Christians in Macedonia to share in their earthly blessings and give money for the poor in Jerusalem (Romans 15:25-26). He encouraged the Corinthian brethren to give to the harvest, especially to the workers in the harvest field (2 Corinthians 9:10). He admonishes the rich to be generous for the Lord's work (1 Timothy 6:17-19). However, we do not see a specific description that Paul asked people for money for his missionary support. We can only infer this.
I think, the idea of being sent by the Church in Antioch also means the concept of fund-raising. We do not see Paul going around raising his own support. We can assume, however, that Paul was being supported financially by this church. Whether he went around himself or someone else was doing this for him, we cannot be certain. One thing for sure is that the churches that Paul worked with were generous in their giving and supportive of missionary work. (Cf. 2 Corinthians 8:1-5) This is probably one of the secrets of Paul's success in missionary work.