Given that the Zoltar machine was quite 'retro' how about a circuit to match. Not even an IC in sight, just a few transistors and a couple of relays.
Scary at first if you can't read a schematic it is the complete machine - push button to start, motor (M1) turns the head AND music plays (MP3) and when it finishes a card will be dispensed driven by motor M2. Just to make certain you can't start it up again until the card is dispensed the 'start' button is disabled.
Let's break it down into smaller (circuit) parts. You can build and test each part
The start trigger and first timer.
The circuit shows the on/off switch (SW1), the 'trigger' circuit is made up from Q1 - a pnp type general purpose transistor, R1 which limits the base current to Q1 - say 10k) and a push to make switch - SW2. The base current must also pass through the second relay switch contacts but for now we will assume these are closed. When the SW2 is pushed Q1 is turned ON and current flows through Q1 - most goes through R2 to charge up a large capacitor (C1), some goes through R3 (330R) to turn ON Q2 (an NPN transistor - say 2N2222) through R3 (2K2). When this transistor is turned ON a current flows through relay 1 coil and operates the relay's switches. D1 (1N4001 or similar) is there to protect the transistor when the relay switches OFF. When SW2 is released Q1 switches OFF but C1 is now charged (up to 12V) and begins to discharge through R2 and R3 into the base of Q2 keeping relay 1 turned ON for a short time. (Enough time to get the head moving and play music).
Relay 1 switches
When Relay 1 coil is energised the switches put power (12V) onto motor 1 (which turns the head, the MP3 player - which plays the music AND through D3 (1n4001) and R4 (330R) it charges up C2. At the same time the second switch contacts DISCONNECT next stage. When Relay 1 turns OFF the second set of switch contacts connects the now charged capacitor (C2) to connect to the final stage.
Final Stage (relay 2)
After relay 1 has been turned ON and then turns OFF the charge in C2 an turn on Q3 (an NPN transistor e.g. 2N2222) through R5 (say 2k2). The first set of relay switch contacts will operate a second motor (M2) which you can use to put out a card. At the same time the second set of relay switch contacts opens the circuit between Q1 and the push to make switch PREVENTING someone from pressing the start button again BEFORE the whole sequence is finished. Once the capacitor (C2) is discharged Q3 turns OFF and the relay turns OFF. The machine is now ready for the next customer.
Component values are not particularly critical C1,R2,R3 determine first ON time. C2, R5 determine second timing (you may to to increase or decrease values to adjust timing). All diodes are 1N4001 types (or similar), Q2 and Q3 are NPN general purpose such as 2N2222 (enough collector current to drive the relay coil) and Q1 is just about any PNP general purpose type such as BC577. All resistors are 0.25W 5% types. Relays are 12V Double Pole Double Throw types (DPDT). Have fun.