Unfortunately, I have had this happen many times in regard to lease applications and other situations. After reaching out to the agent a few times via multiple contact methods, I will send another email and cc: the broker/supervising agent with the hopes that it might prompt them to respond. I've found this to be successful at least 50% of the time.
Each agent is different and operates under their own schedule and operations. You should expect to wait at least 24 hours for a response and your follow up. If it remains unanswered after more than 3 business days (does not include weekends), you may also attempt to contact the broker directly. If no response, you may consider informing your client to search for a new property. If a payment is involved, you can contact TREC Hotline for legal advice. It is always best to refrain from any negative or unprofessional messages despite the frustrating situation. We never know what someone may be experiencing in their lives/career and is always best to remain calm and professional.
In my experience, I always call the agent prior to submitting any applications so that we can alter the application based on multiple applications etc. If they do not answer the phone call, I will send them a text message and try to give them a call either later in the day or the following day depending on the time sensitivity. If I still do not hear back from any method of communication, I will call the broker!