The Liberation of Terra Vol I, speaks to this a little. After taking the Rim Worlds republic, Kerensky put General DeChaviler in charge of using the Rim Worlds industry that was already producing material for war, and used that to rebuild and rearm the SLDF. About this same time, large numbers of house troops started to defect to the SLDF, brining mechs, tanks, dropships, and even two warships (some with house lord encouragement and some without, even though publicly they all disavowed any support).
The Star League State Administration offices also tried to keep up tax collection, and while not very successful did keep some money coming in for the SLDF and other services. Also the SLDF was able to trade technology secrets for parts and funding, as well as "offering lucrative future contract pricing". When that was not enough, the SLDF stripped every base they had and sold anything not needed for the war effort, at greatly reduced pricing.
They also would "liberate" house supply depos to "collect back taxes". Some of these depos left for the SLDF, but had to be "raided" so the house lord did not not appear to be given support, and some were actually raided.
