Maybe. Certainly in the past, before the world was aware LLMs on the level of ChatGPT were possible with today's technology. OpenAI's chosen not to release any real details about GPT-4, so we don't actually know what it would take to train a model of equivalent quality, especially considering training isn't a one-shot. Multiple training runs easily add up training costs. So training for a 12-figure parameter size model(s) (175B) is assumed to be very expensive. But there has been great progress made for optimized models which are smaller by a two orders of magnitude - 7B for a debatable drop in quality (7B alpaca is in no-way competitive with ChatGPT, but it's still very much not a markov chain from during the AI winter). So one possibility is that OpenAI chose not to release salient GPT-4 details is due to it being much smaller than GPT-3's 175B model size and they're hiding the details because of how much that cuts down on training costs. (Which I should note is unsubstantiated conjecture but not outside the realm of possibility.)
The other aspect is that fine-tuning an existing model is way cheaper than creating a competing model from scratch, so a company could offer CompetitorGPT/CompetitorCoPilot competitive with GPT-3.5, and offer fine-tuning of that model trained on the source code repository of the purchaser company's codebase, possibly on-prem or at least inside their AWS VPC/Azure/GCP equivalent.
The other thing to note is that OpenAI is hosting ChatGPT as a public resource available to anyone with an account, akin to Google being open to the public from day one (although that is without an account. Maybe Gmail is a better comparison). I can't say for certain, only OpenAI would know for sure, but I'm willing to bet that inference for ChatGPT is the vast majority of their costs (which is all but trivial). Any private internal-only instance of OpenChatGPT (using the unlicensed leaked LLaMA model or a legal copy or someone else's) could be paying (relatively) minuscule training costs, and way lower inference costs if it's internal-use only. Whether that cost can be borne by a small SaaS company's existing AWS budget is up in the air, which is to say ultimately that you're right - ChatGPT would be difficult without the support of Microsoft via a huge Azure grant, it's less obvious that a self hosted internal-only OpenChatGPT, not from OpenAI, would be possible by hobbyist self-hosters with a prosumer GPU cluster (Say with last generation K80's instead of business-priced A100's), or by a company wanting to leverage LLMs for private use by that company that wants to provide a Copilot like productivity multiplier internal tool to their developers, without sending private source code to OpenAI in lieu of a privacy agreement with them.