First of all, you should contact your insurance company which will tell you if you need to be bonded or not. The reason is that it is not the requirement of all companies to be bonded. However, business insurance is the best solution for any independent contractor in this regard.
Make sure that you keep a good record of your previous work and also avoid any sort of corruption. Keep your record clean as those companies who provide surety bonds will see your criminal history and can reject your request if you have a bad record.
Also try to maintain your financial record and present it to the company that issues bonds if they request you to show your history.
Now you should consult with the licensing and regulations department to verify the requirements of the bond. You will easily become bonded if your profession needs a state license and you are fully eligible for that.
Now if your client wants a surety bond from you, then call your insurance agent and request for a surety bond.
However, if you are an American and do not get the bond locally from your state, you should contact to the U.S. Small Business Administration (SBA). The SBA issues surety bond that cover 70 to 90 percent of the surety risk of the independent contractor.