The key is whether the writer wants to emphasise that the process is ongoing
In general you are right:
"We sold 200 cars last year."
"200 people joined our club last year."
"200 people visited the shrine last week."
However when speaking of a period that goes up to the present moment, where nothing special happens at this instant in time (other than the writer having to quote figures up to now), writers typically use the "have" version. The implication is that the process is ongoing and this is just a progress report so far.
"We have sold 20 of the new style of car since it first went on sale in January."
"200 people have joined our club."
"200 people have visited the shrine."
When writing a note of thanks, one is typically emphasising how soon one is writing, to show that one is prompt in expressing appreciation. Moreover, one may be still be expecting more help in the time to come.
Similarly the writer may be wishing for, or requesting, a continuing stream of events. Hence:
"Lots of people have helped us in this first difficult month with the new baby."
"Lots of you joined our club last month. Will anyone who hasn't joined, please do so?"
"200 people have visited the shrine already. When it appears on Google Maps, many more will come."
I agree with the exam preparation book that the best answer is "have helped us", but I do not think your answer is terrible or would be considered incorrect by most English speakers. They have unfortunately made it difficult for you to instantly pick "have helped", by saying "last year", which doesn't particularly strongly suggest an ongoing process.
More idiomatic UK English would be "have helped us over the past year". This underlines the ongoing nature of the need for help, and that the slice of just one year is arbitrary. "Keep the help coming, please!"