OP - I understand your SIL is upset because her children were upset.
I understand your SIL and her family have no idea about lambing season and that if stressed they miscarry (as I didn't till MN either).
However, as upset as your SIL is, is only a portion of how upset the farmer is/was. The farmer perhaps reacted poorly, perhaps, but she was upset because her flock was in danger. Her livelihood was in danger. To you and your SIL they are "just" sheep but to her they are much more.
Telling the farmer (in your letter) that she needs to rethink how she approaches people will not do any good. It will only have the same reaction your sister is having, except the farmer is vilified as it was YOUR family that is in the wrong here and created the situation. Sending an apology is a great idea, sans the "you should change your attitude" paragraph.
It doesn't matter that the sheep were lambing, your SIL and her children abused animals. Sure, they didn't realise what they were doing was abuse, but it was. It could have (and may have, you don't know yet) resulted in death. Both of the lambs and the Ewes.
The walk way doesn't mean that anything on that through way is yours or the public's property. It doesn't mean anything on that property is up for grabs. It doesn't matter that the kids "thought they were playing a game", it matters that that game involved someone elses property. Sheep are cute and all but they're still private property. Just because I have a shiny car doesn't mean your SILs kids can take it for a joy ride for "a game".
I must ask though. Why is it that the absence of the sign means they can do whatever they want with someone elses property? Why must they put up a sign telling people not to do something that most sensible people don't need a sign for? Why is a sign needed to tell people what should be common sense - "it's not mine, so I need to stay away"?
As an aside, perhaps you could suggest to the town council that they release some sort of education brochure or program for the area... you may find though that many people will think it's pretty bloody obvious so focus on "townies" that don't know. Perhaps your SIL could do the same with her children's school, especially as they "townies" are the ones that need to learn.
Honestly I do understand why you and your SIL etc are upset, but they are being totally unreasonable to not understand that what happened is purely of their own doing. THEY put their children in a position to be frightened. Their actions (or inaction rather) are to blame here. Their lack of common sense is to blame. Not the farmer, who was understandably upset for an unprovoked attack on her property and livelihood. Ignorance is not an excuse.