Can I use existing programs written to control the 4155 and 4156 to also control the B1500A?

The answer to this depends on the model (A, B, or C) of the 4155 or 4156 that you are using and on the command syntax that you are using. The B1500A does not support the SCPI command syntax, so programs that use this command syntax to talk to the 4155A/B/C and 4156A/B/C require a complete rewrite. However, a VXIplug&play driver is available for the B1500A to make this process easier. Programs written for the 4155B/C or 4156B/C that use the Keysight FLEX command set can be ported over to the B1500A with only minor modification, assuming that the B1500A hardware configuration is compatible with the 4155B/C or 4156B/C hardware configuration.