Is there a Pythagorean triple whose angles are 90, 45, and 45 degrees?

You cannot have an integer Pythagorean Triple whose angles are $45°, 45°$ and $90°$.

Assume on the triangle we have sides $a$. Then by Pythagoras' Theorem,

$$a^2+a^2=2a^2=(a\sqrt{2})^2$$

This means the hypotenuse is no longer an integer length, because now it measures $a\sqrt2$. This means no such Pythagoren Triple exists.

No, since if the perpendicular sides are $a$ in length, the hypotenuse would be $a\sqrt2$. But $\sqrt2$ is irrational, so $a\sqrt2$ is not an integer.

In your context you might be interested in isosceles triangles that are almost right. As others have said, a right isosceles triangle has sides that are $a,a,a\sqrt 2$ and as $\sqrt 2$ is not rational we cannot have an integer sided one. However, if we find a rational number that is close to $\sqrt 2$ we can find isosceles triangles that are close to right. We have $\sqrt 2 \approx 1.414213$, while $\frac 75 = 1.4$ is not so far away, so a $5,5,7$ triangle is close to right. In fact the angle is $\arccos \left(\frac 1{50}\right)\approx 88.85^\circ$. You might have enough give to tolerate that. If not, given one triangle in the list is $a,a,b$ the next is $a+b,a+b,2a+b$, so the next is $12,12,17$, then $29,29,41$, and so on. The get closer and closer to right as you progress. If you are interested in where this comes from, you could look up Pell's equation.