I'm not sure I am convinced by reward systems for what is essentially homework in Y2, and thus usually outside of the children's control.
IMO ideally children should read for fun, not for stickers. Putting an external incentive on reading can contribute to reading becoming a chore.
Ok so some children in Y2 should ideally read more, because they need more practice, even if they don't enjoy it. They need to read more so they develop to the stage where they can start reading for fun. But IMO, if reward systems are your method of choice, these children need an individually targeted reward/incentive system, not a class or school wide one.
With any incentive system you have to think carefully about what you want to achieve, and who you need to target. The children or the parents? Those who already read as often as they can (which may be once or 5 times per week), or those who can't be bothered?
So if you want to increase how many times/week children read at home, you need to figure out why children are not reading at home (enough). For example, in my DS' class, which is very middle class with very engaged parents, the majority of children read daily or nearly daily. Those that don't are those whose parents are in the middle of splitting up, those who have recently had a new baby, those whose parents both work full time with the child in before- and after school club from 8-6 every day, those who hate reading and whose parents have decided to take the pressure off, those who have a parent with mental health issues.There are perhaps two children where the parents seem uninterested in what the child does at school.
So what does a system that rewards how many times a child reads at home achieve in circumstances such as ours? It won't make anyone read a single book more, because those who read lots already, can't read even more, and those who don't, have reasons for it that won't change just because there is a reward. With maybe one or two exceptions.
It will however have 'side effects' - every incentive system does. In my example it may cause those children who aren't reading at home daily to feel even worse about themselves, or about school, than they are probably already feeling. Because they will see their friends receive rewards for things that are, for them, unachievable, and out of their control.
Your aim may not be to increase the frequency of home reading. It may simply be to show those children who do read at home, that you value and appreciate that - which is fair enough. In which case, just keep in mind what message you are giving the other children at the same time.