Can I legally play battle actions that increase the current province strength, such as indomitable home and siege volley at a battlefield with no province, such as with ambush?
province strength boosts at non province battles
A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
Can I legally play battle actions that increase the current province strength, such as indomitable home and siege volley at a battlefield with no province, such as with ambush?
In those two cases, Indomitable Home and Seige Volley, yes you may take those actions during a battle at a battlefield with no associated province.
There is nothing in their constraints restricting or requiring the battle to be at a battlefield at a province, or targeting a province. So, when you take those actions, the change to the province's strength simply fail because there isn't one.
Now, there may be other cards with actions that have such restrictions, and in those cases you couldn't. But I hope this answers your question sufficiently.