IANAL, but it is an interesting question to consider whether it would be illegal in Australia (if anything, as a test to see if the right laws are on the books to block this kind of thing). The laws are likely different in the US, and it might vary from state to state.

The Fair Work Act 2009 (Commonwealth), s325 provides that:

An employer must not directly or indirectly require an employee to spend, or pay to the employer or another person, an amount of the employee’s money or the whole or any part of an amount payable to the employee in relation to the performance of work, if:

(a) the requirement is unreasonable in the circumstances; and

(b) for a payment—the payment is directly or indirectly for the benefit of the employer or a party related to the employer.

I think you could imagine the employer arguing a few lines:

• The employee is not required to spend, it is only a factor in promotions and not retaining the same role. OP said you can “get in trouble for not using this” - countering this defence perhaps depends on proving what kind of trouble to show it is a requirement. In addition, under s340, employers are not allowed to take an adverse action against an employee for exercising or proposing to exercise a workplace right, and adverse action includes discriminating between and employee and other employees of the employer.
• That the employee is not required to pay any particular person, they can choose what to buy as long as the select from a prescribed list. However, I think that could be countered by saying this is an indirect requirement to spend, and the “or another person” attaches to the “pay” part, so I don’t think that argument would fly.
• The the requirement is reasonable - however, that could be countered by arguing the privacy angle, and the fact that this is for personal shopping, far outside the reasonable scope of an employment relationship.
• That the payment isn’t for the benefit of the employer. I think that could be countered firstly by arguing this is a requirement to spend not pay, and event if it was to pay, it is indirectly for the employer’s benefit since it allows them to attract and retain clients. The way they are pushing it could further prove this.

So I think it would probably be contrary to s325 of the Fair Work Act in Australia.

Another angle could be the right to disconnect under s333M of the Fair Work Act:

An employee may refuse to monitor, read or respond to contact, or attempted contact, from an employer outside of the employee’s working hours unless the refusal is unreasonable.

If someone has a work and a personal phone, and has the app on the work phone, but refuses to use take the work phone or install an app on their personal phone so they can respond to tracking requests from the employer, then maybe this also fits.

I also wonder if in Australia this could also be a form of cartel conduct - it is an arrangement of where purchases (other than those the company should legitimately control) are directed centrally under an arrangement by an organisation.

Under s45AD of the Competition and Consumer Act 2010,

(1) For the purposes of this Act, a provision of a contract, arrangement or understanding is a cartel provision if: (a) either of the following conditions is satisfied in relation to the provision: (i) the purpose/effect condition set out in subsection (2); (ii) the purpose condition set out in subsection (3); and (b) the competition condition set out in subsection (4) is satisfied in relation to the provision.

So the purpose condition has several alternatives separated by ‘or’, one of which is:

(3) The purpose condition is satisfied if the provision has the purpose of directly or indirectly: … (b) allocating between any or all of the parties to the contract, arrangement or understanding: (ii) the persons or classes of persons who have supplied, or who are likely to supply, goods or services to any or all of the parties to the contract, arrangement or understanding; or

It sounds like there is a solid argument the purpose condition is met - they are allocating where people who are part of the arrangement (employees) shop.

They’d also need to meet the competition condition for it to be cartel conduct. For this to be met, the arrangement might need to include the clients of the company:

(4) The competition condition is satisfied if at least 2 of the parties to the contract, arrangement or understanding: (a) are or are likely to be; or (b) but for any contract, arrangement or understanding, would be or would be likely to be; in competition with each other in relation to: … © if paragraph (2)© or (3)(b) applies in relation to a supply, or likely supply, of goods or services—the supply of those goods or services in trade or commerce; or

So it could be argued that this is a cartel arrangement between the company, its clients, and its employees, and so attract penalties for cartel conduct.