Can I list family business and informal work on my CV if there's no documentation

I graduated with a BA in Economics in 2024. Since then, I completed a Data Science certification and prepared for my Master's entrance exams. I didn't get into the college I wanted, so I've decided to wait instead of enrolling somewhere I'm not happy with.

I've been job hunting for almost a year, but I think my lack of formal experience is holding me back.

During this time, I've been helping run my father's business by overseeing day-to-day operations, coordinating workers, and making sure work gets completed. I also did part-time work for a logistics company, but it was an informal arrangement with no offer letter, contract, or other paperwork.

Can I honestly include these experiences on my CV? Will recruiters consider them legitimate experience? And if I get hired, how do background checks work when there's no official employment documentation?

I'd really appreciate any advice, especially from recruiters or hiring managers. Thanks!