You could and I have heard this, but it would be better to use the correct adjective: "sunny" to describe the day, e.g: "It's really sunny today." The reason is that "sun" is not a measurable quantity. A "sun" is a countable noun, and for the same reason you would never say "There is a lot of dog", instead "There are a lot of dogs." "There is a lot of water" is OK, because we measure water. One can measure "sunlight", so theoretically "There is a lot of sunlight" is correct, but it sounds unnatural unless you are describing something like the interior of a house where people normally check how much sunlight comes through windows ("Ah, this room has a lot of [sun]light!"). Sometimes people use the word "sun" as if it were an uncountable noun meaning "sunlight", and that's when you may hear "There is a lot of sun". This arguably is substandard language, but it wouldn't be difficult to understand.

There is a lot of sunshine today.

not really, "you can say its really hot" or "Its sunny today" or "the sun is really shining"

In your question you do not want to think of "sun" as a countable noun. You're not asking how many suns there are. You're asking about the quality or quantity of the sunlight. 'There is a lot of sun today' is fine. But it would be better to say, and I would suggest you say It's very sunny today. Or something of that nature.