Can the Soul society exist without the soul king?

The Soul King by himself is not the key to Soul Society's survival.

It is true that someone of transcendent power needs to act as the linchpin holding the three dimensions (Earth, Soul Society and Hueco Mundo) together. However, that person needs not to be the Soul King himself.

The requirements to become a Soul King are to possess reiatsu from the two spiritually-attuned species (Shinigami and Hollow) combined with a fragment of the Soul King. Alternatively, possessing an organ of the Soul King may also make one a viable candidate for the title of Soul King, even if that person isn't both shinigami and Hollow.

It should be noted that each and every Fullbringer possesses a fragment of the Soul King ; and that their Fullbringer powers derive from that fragment. Even though it is explained within the manga that Fullbringers gain their powers when their pregnant mother is attacked by a Hollow and survives (as explained in Chapter 433, page 10), that explanation apparently was incorrect. The Hollows attacked the mother BECAUSE their offspring bore a Soul King fragment.

Several characters have been noted (including within the canon-ish Can't Fear Your Own World light novel) to be Soul King candidates :

1. Yhwach : Since he absorbed the Soul King's entire power, he became a suitable candidate for the very title of Soul King himself.
2. Ichigo : Notably possesses both Hollow and shinigami reiatsu. He also mastered Fullbring, thus proving he possesses a Soul King fragment.
3. Kūgo Ginjō : Former shinigami and proficient Fullbring user. Upon stealing Ichigo's Fullbring, he also copied his reiatsu structure, thus gaining Hollow properties, as explained by the Bleach wiki : "Upon absorption of one's Fullbring, he also incorporates their own Reiatsu into the attack, making it virtually identical to the original user." (Chapter 470, page 8)
4. Jūshirō Ukitake : Was used as a host body by Mimihagi, the Soul King's chopped-off, conscious right hand. By sacrificing himself and unleashing Mimihagi, he could have taken, theoretically, the Soul King's place (although his attempt thwarted by Yhwach).
5. Hikone Ubuginu : A character exclusively developed within the light novel Can't Fear Your Own World. In short, an artificial individual built from shinigami, Hollow and Soul King fragments to achieve transcendent power and take the place of the Soul King.

The Can't Fear Your Own World light novel implies that a high amount of spiritual pressure is needed to be a Soul King candidate, although since we are lacking an official translation, I cannot vouch for this information's truthfulness. Likewise, this light novel shows that a defeated Yhwach has been sealed and now replaces the Soul King. This information is most likely canonical. If Yhwach had won, it is very strongly implied within the manga (and confirmed in the CFYOW light novel) that Ichigo was to be forced to take the Soul King's place.

TL;DR : The three dimensions' structural integrity can be preserved if a Soul King candidate is sacrificed to become the linchpin in stead of the original Soul King.