Although I would collect something for my friends or family, it really is only polite to ask first. You never really know what someone else has going on in their life, whether they live five minutes away or five hours away.
You managed to ring her and speak to her to ask her to collect it after you had purchased it, so surely you could have done so before you bought it.
And you say "waiting for SIL to answer the phone or return an email can take time" even though she has no children and works part time. But it does imply she still has a busy life, which you have made a demand on without asking first if she would mind.
It really makes no difference what you have done for her parents. They are also your DH's parents and your in-laws, you have your own relationship with them, and it's your DH more than anyone who should be pleased you have a good relationship with them and want to help them. I would be pleased that my SIL helped my parents, but I wouldn't want to think it made me beholden to her in some way.
And you say this "I asked her to give a day and time of her own choice to collect" but it comes across a little like you were doing her a favour in letting her choose the day and time. It was already a done deal, the clarinet was bought, so not really that much of a choice for her to pick when she collected it.
I don't think she is being unreasonable. She has agreed to collect the clarinet and bring it with her to deliver it to you at Christmas. She has also asked that in future you ask first if you want a favour like this and that seems fair enough to me. She is still helping you out now and hasn't refused to help in the future. She's just asked for a little consideration first.
I know it's not a problem to you but there are many reasons why it might be to her and her feelings are still valid even if you would feel differently or would do it for her.
Perhaps she already feels put upon from someone else and this has been a last straw moment, perhaps she's having an especially busy couple of weeks (as we all seem to have in the run-up to Christmas, even us part-timers 
).
Perhaps she's not feeling well, perhaps she's worried about the responsibility of collecting the instrument in case it is not as described or she gets the blame for it being damaged in some way.
Perhaps she can't afford the petrol to drive and collect it, or she's wondering how to fit it in the car for the visit at Christmas when she and her DH will have luggage and presents etc, and perhaps their own instruments to squeeze in as well.
Or perhaps she's just someone who prefers to be asked first and is confident enough to say how she feels when that doesn't happen. I don't think that's unreasonable, she sounds like she did so politely and she is still collecting the clarinet and doing you the favour, she's just asked you to ask first in future.