Suppose all the pieces are on the board. Does there exist a position such that
A) one of the players can't make any move (a stalemate)?
B) neither player can make any move (a double stalemate)?
Disclaimer: This solution is not reachable from the starting position, and is not reachable in a game of Chess960 (thanks Rewan!).
[FEN "3bBNRN/2pPpPKQ/2P1P1PR/7P/p7/rp1p1p2/qkpPpP2/nrnbB3 w - - 0 1"]
Why does the solution here not work?
This is clearly not reachable from the starting position (because of the bishops stuck on the first rank), but the question does not state the position must be legal.
Here is an example of a 12-move game after which White (to move) is stalemated. All 32 units (pieces and pawns) are still on the board. The original version of this concept game was created by Charles Henry Wheeler, and published in Sunny South in 1887, according to Edward Winter's C.N. 3679. Samuel Loyd is often, and wrongly, given credit.
[Title ""] [StartFlipped "0"] [fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.d4 e5 2.Qd2 e4 3.Qf4 f5 4.h3 Bb4+ 5.Nd2 d6 6.Qh2 Be6 7.a4 Qh4 8.Ra3 c5 9.Rg3 f4 10.f3 Bb3 11.d5 Ba5 12.c4 e3 1/2-1/2
To answer the second part of your question:
Suppose all figures are on the board. Does there exist a transposition of figures such that both of the opponents can't do any move (a stalemate)?
No, this is not possible. Pieces are simply too mobile for this, so for a stalemate you need to hem them in (like the white queen in @RosieF's answer) or pin them on the king. You cannot use pins for a double stalemate, since that would imply the pinning piece can move and it's not stalemate for the other player. (That piece could be pinned itself, but there's no way to make a circular pin.)
The question has two parts to it, and Rosie F perfectly answers the first part. Regarding only the second part, the question asks for any possible position where all 32 pieces are stalemated, disregarding legality of such positions. @im_so_meta_even_this_acronym's wonderful answer proves that it indeed possible. However, I want to focus specifically on only legal positions just for fun. I thank @TonyK and many others for helping contribute to the quality of my answer.
Speaking specifically about legal positions the best possible answer is that 32 pieces cannot be stalemated. The known maximum number of pieces that can be legally stalemated is 30. I have two examples. Remember, these are just records, with the ability to be improved upon. Please do try to understand that I am not contradicting any other answers-I am delving into a single area that no other answer covers (i.e. I am speaking of only mutual legal stalemates. Illegal positions with 32 pieces have been proven to exist in another answer.)
I found the below first position here.
[Title "30 Stalemated Pieces, Gustavus Charles Reichhelm Brentano's Monthly 01/1882"] [FEN "rqrb4/nbk1p3/p1p1P3/PpP3p1/1P3pPp/3p1P1P/3P1KBN/4BRQR w - - 0 1"]
@Laska also found another legal position that has 30 stalemated pieces, with a promoted piece.
[FEN "brnbKRRN/qnk1pBN1/rb1pPpPp/p1pP1P1P/PpP5/1P6/8/8 w - - 0 1"]
I do suppose that it would need to proven that 30 it is the maximum under the particular condition that I have chosen. I have two ""semi-proofs" that illustrate that it just might be the limit.
Firstly, all pawns must be spread across the board, facing each other through the 2nd to 7th ranks. The only way to really block pieces in is to use the king, and the pawn arrangements that can be used to block the king and the pieces is very limited. The 1882 position does this extremely well.
Take a look at the two following positions.
[Title "Edgar Fielder, British Chess Magazine, 1938, Only 1 Out Of 32 Pieces Can Move"] [FEN "brn1BRRB/qknQpKNp/rb1pPp1N/p1pP1PpP/P1P3P1/1p6/1P6/8 w - - 0 1"]
[Title "Thomas R. Dawson, The Chess Amateur 1923, Only Two Possible Moves For All 32 Pieces"] [FEN "qrrb2n1/nbk1p3/p1p1Pp1p/PpP2P1P/1P1p4/BK1P2p1/QRB3P1/NRN5 w - - 0 1"]
These two problems show just how constrained possible pawn structures can be. Finding one that prevents all possible moves is a tough task. The 1882 position does this very nicely, with only two knights left out of the fun. While it is not definitive, it almost certainly shows that 30 is the limit for a legal position.
Secondly, if you want an authoritative source for 30 for the record, as late as 2013, even Anatoly Karpov gave the the 1882 as the record in his book Finding The Right Plan.
He does give this nice proof game for it.
[FEN ""] 1. Nf3 Nc6 2. Nc3 Nf6 3. Nb5 Ng4 4. h3 a6 5. Na7 Nh2 6. Rxh2 Rxa7 7. g4 b5 8. Bg2 Bb7 9. e4 d5 10. Ke2 Kd7 11. Qg1 Qb8 12. b4 g5 13. Bb2 Bg7 14. Rf1 Rc8 15. Bd4 Be5 16. Rh1 Ra8 17. Nh2 Na7 18. f3 c6 19. Bf2 Bc7 20. Be1 Bd8 21. Kf2 Kc7 22. a4 h5 23. a5 h4 24. c4 f5 25. c5 f4 26. e5 d4 27. e6 d3
Feel free to tell my of any more "semi-proof/proof" that can be added to this list.