Bug sprays aren't an issue really - just exclude the puppy from the garden while you are using them, and allow him access again once the spray has dried.
What can be a problem with pets is granular lawn treatments - best not use those with pets.

Another factor would be exactly what kind of bugs you want to kill. Many of the safer organic remedies aren't very "broad spectrum", and have little residual action, so you can't just spray once a month, just in case there is a problem. Probably just as well, since that sort of treatment was killing the environment and breeding resistant strains of pests.