I agree with PPs that sending the letter would be a mistake.
I also agree that you don't need to participate. My response would depend on how the approaches are made:
- general e-mail to all staff - ignore, unless it is an official request
- direct mail or personal request - polite, friendly "no thanks"
- collection for a birthday or other life event - I would try to contribute if you can - fitting in with the culture of the workplace is usually a good idea. - if the person making the collection is requesting a specific amount and this is too expensive, then it might be worth raising it informally with co-workers to mention you're finding it's pretty expensive, and suggest perhaps a cap on how much is suqgested, or a change to a 'give what you want'.
You don't have to give any explanation for not participating, but if you are asked, and feel that some explanation is needed or the sake of good relations with co-workers I would stick to something vague "It's not in my budget at present" or "I'm already committed to supporting my daughter/neighbour/neice's troop" or something similar - that way, you are not coming across as critical of the individual collecting/selling, or of the culture of asking at work, but you can chose not to contribute to things you don't want to.
I would not involve HR unless you find yourself in a situation where you are being bullied or pressured if you don't join in. It seems like over kill to go straight to HR without first seeing whether a polite 'no thanks' will work.
In the case of the person you bought something from, I would send them a mail setting out what you ordered, when you ordered it, and when you understood you would be receiving your items. Ask them to confirm when you will get your items and then chase them again on that date if they haven't arrived. If the delay is excessive, you could request a refund instead.
If nothing happens, check the office manual (if there is one) to see what policies there are. Depending on what it says, you could ask HR to mediate between you and the co-worker who sold you the stuff to sort out a refund or timescale for your good to be provided, but unless she was selling on behalf of the company this would appear to be a private matter between you and her, not the company's responsibility to resolve.
If selling at work is against the company policy then you can report it to HR as a breach of policy but you may also be in breach for having bought from her, so check the policy first.
In my office, it's very common for people to do sponsored events, or to let everyone know that their child is doing a sponsored event, or to offer things for sale, or in return for a charity donation and there are always collections for life events such as the birth or adoption of a child, retirement, weddings and milestone birthdays - but these are all completely voluntary.Most people will contribute to some things bit not necessarily to everything, and no-one is pressured or 'chased' to contribute.