Yes, you can run your air conditioner (AC) and use a vacuum at the same time on a hot day. Both of these appliances use electricity, and running them simultaneously won't cause any issues as long as your electrical system is in good condition and can handle the load.
Modern homes are typically equipped with electrical systems that are designed to handle multiple appliances running simultaneously. However, if you have an older electrical system or you're using other high-power-consuming devices simultaneously, it's essential to be mindful of your overall electricity usage.
Using both the AC and a vacuum cleaner shouldn't cause any problems, but keep in mind that running multiple high-power appliances at the same time may lead to a higher electricity bill. Additionally, if your electrical system is outdated or prone to tripping breakers, running too many appliances simultaneously could cause an overload and trip the circuit breaker. If you notice any issues or unusual behavior with your electrical system, it's best to contact a qualified electrician to assess and make any necessary improvements.