This phrase is a modification of the crude, informal present-day phrase, "You can stick it up your ass."
This phrase indicates two things: The person speaking does not care about it, and the person speaking does not like the person who is being targeted in the phrase. "Carry it with oneself always" is a good guess, but this phrase doesn't have that meaning at all.
The use of "They can stick my application form up their jumpers" is used to say that the not only does the speaker not like the people who have his application, and he also doesn't care about the application. "They can stick my application form up their asses" would have been about the same thing, except that in the phrasing the speaker used, he's probably making fun of them for wearing jumpers, too.
This phrase is very rude and not a very good "joking" insult -- it should only be used when you really don't like someone!
The humor comes from the fact that the speaker has used this phrase to declare his contempt for the people involved in a round-about way. I wouldn't say it deserves bursting into laughter, but there you go.
Meaning
See you in the funny paper[s] means "Goodbye, see you soon".
A Dictionary of Catch Phrases (1986) by Eric Partridge and Paul Beale says:
see you in the funny papers (—often and orig. I'll). 'This jocular farewell suggests that the person addressed is rather laughable: US: 1920s; extinct by the 1950s' (R.C., 1978). Perhaps adopted in the UK from American servicemen c. 1943. By c. 1955, (I'll) see you in the funnies.
Etymology
The OED has funny paper from 1874 and funny column from 1860, meaning "a (section of a) newspaper containing humorous matter or illustrations".
1920
The earliest example I found of the phrase is in a letter in Commercial Telegraphers' Journal (August 1920, Vol. XVIII, No. 8):
So long, boys, see you in the funny paper. "30."
J. N. HANNA,
Box 1004.
1921
Another in the Union Postal Clerk (March 1921, Vol. XCII, No. 3):
We will see you in the "funny paper" next month.
PRESS COMMITTEE.
1921
Here's a April 15, 1921 letter published in University of Virginia student paper The Virginia Reel (April 18, 1921 Vol. 1, No. 8):
Well, boys, must close now. La, la, till the next time, and I'll see you in the funny papers. Ever your, ADELAIDE.
1922
Here's a 1922 example in the signoff of a report in The Tusla Scout from Troop 12 by Ed M'Lain (published in The Tulsa Daily World, March 19, 1922):
Good-bye, see you in the funny paper.
This use by scouts suggests it's not insulting, but may be used in a good-natured, light-hearted mocking manner.
Best Answer
'Should' sometimes appear to carry the idea of 'happen by chance''.
If you see Fred tomorrow, please give him my regards. / If you should see Fred tomorrow, please give him my regards.
It's odd/strange/interesting (that) you say that. / It's odd/strange/interesting (that) you should say that.
It was odd/strange/interesting (that) you said that. / It was odd/strange/interesting (that) you should say that / should have said that.
The utterances with 'should' are more tentative than those without.