Excuse my naivety........so were they all bitten by a tick? I have never understood lyme disease and how a person can get it other than getting a tick bite. It seems odd to me that they all got it at the same time.
Ticks are all active at the same time of year, they favor the same sorts of terrain, and they have large nests. So one small area can end up being highly populated, where they'll acquire the same pathogens by feeding on the same animals or from their mutual Mama Tick.
If ticks are common in an area, then people are going to be in contact with them on a frequent basis - maybe even daily, or several times per day, if they spend a lot of time outdoors. Occasionally a tick will stay on long enough to transmit something, and usually an approximate percentage of the ticks in an area will be known to carry certain diseases.
In the countryside, where there are a lot of the natural host(s) which carries the disease, and a lot of human recreational areas, and a lot of tall-ish grass or plants ticks like to ambush from, and where a decent percentage of the ticks carry Lyme or co-infections, I don't think it would be at all odd for several people in the same family to get sick at the same time. That sort of situation is exactly how Lyme was first discovered in Connecticut, when a lot of children started having arthritis symptoms in a close enough time frame to draw suspicion.
It can also take a while for Lyme symptoms to manifest, and they might be mild for a while. So it's possible that one family member picks it up one summer, two more get in the next summer, and a couple more get it a few years down the road. Then someone progresses to severe neurological symptom, they get tested, are positive for Lyme, and the other family members who have also been exposed in the same area get tested too.