Following an unexpected missed approach due to weather; a Citation Excel flight crew with an inoperative autopilot was issued an unpublished Missed Approach Procedure. Their inability to program it in a timely fashion caused them to twice request delay vectors from ATC. Both times the request was refused.

Narrative

On our flight from ZZZ1 to ZZZ2 we encountered some moderate turbulence that rendered our autopilot unusable. On descent we checked the weather in ZZZ2 and it was well above minimums. We flew the approach assigned to us by ATC. Since our autopilot was inoperative and we were experiencing some turbulence we deviated slightly on the approach; however we never exceeded any tolerance. When we arrived at minimums we were unable to see the runway due to the deteriorating ceiling and executed the missed approach while still in IMC. We were given an unpublished missed approach clearance that included a complicated non-published hold at a fix that was very close to our current position. Due to our heavy workload with the autopilot being inoperative; the missed approach and being in IMC conditions we asked for delaying vectors so we could get reorganized; finish our checklist; set up the hold and determine our next course of action. We informed the controller that we had lost our autopilot and needed some time to accomplish these necessary items. The approach controller refused to give us delaying vectors but instead reissued different holding instructions at the same VOR. These holding instructions were the exact opposite of what was published for the missed that we had briefed. The non-flying pilot was having difficulty entering it into the FMS as it would not accept the inputs. Due to the fact we were so close to the fix that we were to hold at; setting up the hold manually was difficult. We finally intercepted the correct inbound course manually. The controls were transferred so the hold could be verified by the other pilot. At this point an incorrect turn into the hold was made. ATC notified us of this and we corrected immediately. Again we requested delaying vectors and again ATC refused to issue these. At this point; it was broadcast that the weather had improved and we were able to land. Contributing factors that lead to this situation was the controller refusal to work with us despite our repeated requests; loss of the autopilot; time constraints; complicated holding instructions; and non-published missed approach instructions. The bottom line was that we repeatedly requested vectors so we could reduce our workload and safely fly the aircraft but were denied.