This question covers a lot of area, so let's work through it piece by piece.
First, the normal approximation for a vibrating string is (a) a transverse (perpendicular to string) displacement that (b) is a small where (c) the string acts like a spring: (small) changes in length result in (small) changes to the tension.
What's "small"? Much less than what's already there.
In that case, there is a periodic change in tension as the string vibrates. It goes like the amplitude squared: two positive peaks and two zeros per cycle of the string. Again, this is small compared to the tension already in the string, and is usually ignored on that basis.
If the amplitude is large, so that the tension change is large, then the motion gets more complicated: Still periodic, but not the nice sinusoidal form with constant frequency. The increased tension at the peaks tends to "flatten" the plus and minus peaks of the sinusoidal motion by pulling back early; it also raises the frequency as the amplitude increases. With even more tension, it gets even more complicated...
But there are string oscillations that behave somewhat differently. For example, a rotary vibration is possible: Think of the motion of a double-dutch jump rope that's circling around with two fixed end points. That's vibrating, but it also has a constant length hence (slightly) increased, but constant, tension.