Yes. It's less likely to cause long term effects than Covid-19 itself based on the current research of studies on those who've caught the virus. The vaccine is being tested for side effects, usually there's quite a long period where peers can review and debate studies, which I think is greatly reduced in timescale. That doesn't mean it won't be assessed by peers or queried, just that the timescale that this happens in is reduced. The vaccine is unlikely to have long term effects. I'd rather risk a vaccine.
Children have died from Covid-19, it's just less common. They have also contracted multi-organ inflammatory syndrome as part of the virus. Equally, 'underlying health condition' can vary greatly in what it means, there are a high number of people with something that would be counted as an 'underlying health condition' from polycystic ovaries to diabetes to asthma to high blood pressure to hypothyroidism to obesity etc. It doesn't have to mean something where the person would be likely to die from it without catching the virus. What I think will be useful later is seeing how many of these conditions did affect recovery from Covid-19 and to what extent, as well as whether any increased the risk of contracting it. At the moment there are suggestions, but further research on this is needed.
Children have been lucky so far, but pathogens mutate and the virus may well be different in the following years to now. This means that the vaccine first given may alter a bit from the first year it is given to the third year or years after (like the flu vaccine). We know the flu vaccine has slightly different reactions each year based on the way it changes and which strains it is aimed at eg. Some years a majority report nausea and a runny nose, other years a minority report that but a majority report their arm aching more than previous years. In a changing vaccine or vaccine expected to change, it makes less sense to wait. We do have a lot of knowledge on creating safe vaccines relating to other viruses (eg. H1N1 vaccine, which was developed quickly) and there has been a lot of investment in the last decade on creating novel development programs that are able to be altered depending on pathogen. Although we can't be sure on long term effects, the majority of long term effects with any possible links at all to vaccines tend towards the minor end of the scale (mild asthma, food allergies). We don't know if a majority of the children who contract Covid-19 will go on to have long term adverse effects if it lies dormant and reoccurs, though it is possible at least some will, as it is thought to pass the blood-brain barrier. However, there does tend to be strong anti-vaccine sentiment which perpetuates the internet and can appear convincing. Often, it is taken out of context. I believe, statistically, you are far more like to have a short term reaction to a vaccine that's dangerous (anaphylaxis - for the 6 in 1 vacc. it's 1 in 100,000) than a long term one - and, even then, the risk of that happening is so minimal that it isn't really a consideration not to have it. Vaccination staff are also aware of this and trained to handle it for the rare cases it occurs. On the whole, I think the vaccine is likely to be safer than the virus and severe adverse long term effects will be unlikely, so I'd rather have the vaccine and make sure my children have the vaccine than not.