The other posters covered this well, if your tractor doesn't have a volt meter, buy or borrow a digital multimeter and check the voltage at the battery with the engine off and then running. Should be around 12-12.5 volts DC off, 13.5-14.8 with it running. You should see a definite increase in voltage as an indicator the alternator and regulator/rectifier (R/R) outputs are OK.
I don't have a sehematic for the Briggs Vanguard to see what type of alternator stator it has, but a typical three phase alternator will feed approximately 30 volts AC between the three legs in to the R/R module. If any one of the fields is dead, the stator is shot and needs to be replaced, which involves removing the engine flywheel. (Yes, a project on a shaft drive Cub) If the feed to the R/R is OK and there is little or no DC charge output from it (indicated by low or no charge voltage), you just need to replace the R/R module, a farily simple task. R/R modules typically fail more frequently than the stator does.
Also, check stupid things like loose cables and connectors. These inexpensive items sometimes cause more grief than they are worth dollar-wise.
-Fordlords-