Our 2 & 1/2 year old granddaughter won't generally let anyone help her do anything she thinks she knows how to do even if it's beyond her ability. Not even her parents, though her parents of course are forced to help her in various situations where she just doesn't have the necessary ability. For example she just wasn't strong enough to click in the safety latch on her car seat so finally after letting her try for a bit, I helped her click it in and she got really mad and cried for about 20 minutes. Is that normal?
My daughter was the same and still is now though once she got older she has better understanding of what she can and cannot do and will accept help when needed. So, first, yes, it is normal. It might be a stage or just the way she is. We chose to encourage our daughter's independence and self sufficiency and even though we had to help her sometimes (which resulted in tears unless we were able to convince her to "do it together" and managed to get it done this way) we took advantage of her want for self sufficiency to teach her to take care of herself, i.e. get dressed, buttons, shoes, feed herself, clean after herself, etc. Now I have a 4 year old who sorts her own laundry, folds her own clean laundry and puts it away, unloads all unbreakables from the dishwasher, carries her dishes to the sink after a meal and wipes the table after she is done eating, makes her own bed, etc. She loves being independent and giving her a lot of "jobs" makes her feel like a big girl and proud that she can do it all herself. It was difficult to teach her it all and do it over after she was done in the beginning, but now she is actually a big help and is happy to do house chores :) So it is not all bad.
Totally normal. I would add to let her do what she wants to do and don't step in unless it is dangerous, or if you are running late because she is physically unable to do whatever it is, ask first if you can help because of whatever the reason is. Check out some Montessori materials, http://www.themontessorinotebook.com/age-appropriate-chores-for-children/. Even toddlers can do a lot and have joy and pride in the process.
I have a 2.5 year daughter as well. My daughter is extremely independent as well. 2 days ago, we had to go through the exact same car seat buckling incident. She cannot do the lower latches but she can do the upper latch in the carseat. It is normal that you put your rules like "I will do the bottom and you can do the top". I don't allow her to do the hard one. The next day after the tantrum, she allowed me to the bottom latches immediately. If you have a rule and be consistent applying the rule, she will be ok after one or two times of tantrum. Independent children should also learn to follow the rules. When I give in more, she wants more and more and my patience level goes lower and lower.
Just have rules, explain to her when her mood is good. Parents and grand-parents should follow the same rules as well. If one person makes an exception, they take advantage of it and start to push the buttons again.