Hi Alex,
When something "expands", it takes up more space or increases in its volume.
When something "extends", it becomes longer than its original length.
Examples: A liquid gains heat and expands while a load hung on a spring causes it to extend.
Generally, "Expand" means to spread out; "Extend" means to stretch out or lengthen something.
Having said so, the difference between these two words is not as simple as the direction of movement of something. (Refer to **)
Both words are also used in other different contexts.
Expand:
1) We can "expand" a business into other sectors. In this case, we are making our business more extensive. (In this context, we are trying to increase the scope of a business.)
2) We may "expand" an investigation or explanation. (In this case, we are trying to elaborate or gather more details on a matter.)
3) A bird which "expands" its wings unfolds them.
Extend:
1) We "extend" the length of our vacation by deciding to stay a few more days. (In this case, we prolong the duration of an activity.)
2) We "extend" our hands to offer someone a handshake. (In this case, we are jutting our hand towards someone to offer a friendly gesture.)
3) We can "extend" the mobile coverage of a city. (In this case, we are stretching out something in all directions.)
** Notice that the word "extend" describes a radial movement instead of a unidirectional movement.
Example: As our business "expands", we will "extend" our operating hours!
Back to your sentence: It needs to "extend" delivery time. (Since we are referring to lengthening the duration of an activity.)
Hope this helps.
Cheers,
Lance