Is the string being stretched/extended only in the direction of the worm?

If so, no. The worm will never get there.

If it's being stretched on both sides, then the worm will travel 1 cm/s and be "set back" only 0.5 cm/s. So it'll get there. It's too early to do the math right now to say how quickly.

It's almost certainly the latter example.
But I suspect it's more that
He walks 1cm, the string is stretched uniformly so all distances are now 50% farther away (1cm onto 2cm) so he has to walk 50% farther leaving him 1.5cm away.
In the 2nd second, he walks another 1cm putting him .5 cm away and we'll add 1cm into 3cm so it'll be 33% farther away or .50*1.333 or .6666cm. So it'll take him 2.666 seconds.


In your case it would take a 3 full seconds.

worm is in middle of string at 1cm. String extends to 3cm total, since worm was in middle it is now at 1.5cm. Moves 1 cm to 2.5cm. String extends to 4cm. Since worm was at 2.5/3 during extension, majority of the extension will happen behind him and he'll make it to the end on his next movement

This part I'm 100% sure about:
As long as there is ever any moment that the worm is walking before the string is being stretched then he will always reach the end eventually as some % of the distance he walked will then be part of the stretched string, and he will not have to walk that and then distance traveled will be > additional distance stretched.

Part I'm only 95% sure about:
He will never make it if the string is being stretched from the moment that he starts walking and it finishes being stretched before the end of the second. The rate that the string is being stretched in front of him will be > than his walking speed so he will not catch up.

Part I'm really not sure about:
The edge case where this string is being stretched over the exact interval that he is walking. I.e. it is being continuously stretched over the 1 second time frame. I think in this case he would stay 100cm from the end at all times.

He always makes it no matter when the string is being stretched because it only ever gains 1cm per second.
And any some time t he will always be not on the starting point so some potion of the stretch will always be behind him. So his 1cm / sec will always be >> % of sting in front of him * 1cm.

He always makes it for any value of string length.

t = seconds
d[t] = distance remaining at time t

d[0] = 100
d[1] = (100-1)+(1-1/100) = 99.99

Yes he will eventually reach the end.

More generically,

m = worm movement per t
s = length of stretch per t

d[0] = initial length of string
d[t] = d[t-1] - m + (s - (d[t-1]-m)/(d[0]+s*t))

m > (s^2+(s+1)*d[0])/(s+d[0]+1)

As you can see, the initial length of the string matters.